Rapid evaluation model for EOR techniques applicability of gas flooding, foam flooding and surfactant flooding based on modified fractional flow theory

Before a wide range of enhanced oil recovery (EOR) techniques were implemented for an oilfield, the EOR potential and economic evaluation of the techniques should be evaluated in advance for each reservoir to determine which EOR technique was proper. In an oilfield developed with fluvial delta reservoirs, the complicated distribution of scattered small reservoirs in vertical and horizontal directions brought trouble for evaluation work. A rapid and reliable evaluation model for EOR techniques applicability was necessary to deal with the evaluation simulation for many small reservoirs of an oilfield. Combining fraction theory model with auxiliary equations, which describe the effect of formation heterogeneity and mechanism of different EOR technique on fractional flow rate, an evaluation analysis method for EOR techniques applicability of gas flooding, foam flooding and surfactant flooding was proposed. In the gas flooding model, the minimum miscible pressure of impure gas was introduced, and the relative permeability was modified by the minimum miscible factor. In the foam flooding model, changes in the mobility ratio and chemical adsorption were considered and a reduction factor of mobility ratio was introduced. In the surfactant flooding model, calculation formulas of viscosity and interfacial tension as well as the relative permeability were introduced. Finally, the model was simulated for a low permeability reservoir, and the simulation results were compared with that from Eclipse software. The similar results, little calculation time and feasibility of predicting optimal injection parameter had shown the reliability of the rapid evaluation model.


A s
The proportion of clay mineral (dimensionless) A rj Parameter-fitting relative permeability curve (dimensionless) A ro Parameter-fitting relative permeability curve (dimensionless) B o Volume factor of oil (m 3 /m 3 ) B s Volume factor of injected solvent (m 3 /m 3 ) B w Volume factor of water (m 3 /m 3 ) B rj Parameter-fitting relative permeability curve (dimensionless) B ro Parameter-fitting relative permeability curve (dimensionless) C foam The volume concentration of foam (dimensionless) C ij Concentration of component i in phase j (mol/L) C j Total concentration of component i (mol/L) C rj Parameter-fitting relative permeability curve (dimensionless) C ro Parameter-fitting relative permeability curve (dimensionless) C surf The effective surfactant concentration (mol/L) C surf,0 The injected surfactant concentration (mol/L) C s The effective foam concentration (dimensionless) C r s The referenced foam concentration (dimensionless) C max OS The surfactant concentration corresponding to the maximum interfacial tension (mol/L) C min OS The surfactant concentration corresponding to the minimum interfacial tension (mol/L) C OS The surfactant instantaneous concentration (mol/L) D grav The coefficient for gravity separation effect (dimensionless) D P The value reflecting inlayer heterogeneity (dimensionless) DS Adsorption ratio (dimensionless) eo The transition index (dimensionless) es The control index (dimensionless) E A The area sweep coefficient (dimensionless) E R Oil recovery (dimensionless) f j Fractional flow of phase j (m s −1 ) f w The weighting coefficient that controls the mobility ratio (dimensionless) F The modified coefficient of relative permeability due to interfacial tension (dimensionless) F ACT The coefficient reflecting the influence of gravity on gas flooding (dimensionless) F STL The contribution of water drive on oil recovery without gas intrusion Certain component in the porous media (i = 1 water; i = 2 crude oil; i = 3 injected gas) k r The oil-phase relative permeability after surfactant injection (dimensionless) k rj Relative permeability of j phase (j = water, gas, injected solvent) (dimensionless) k ro Relative permeability of oil phase (dimensionless) k m r The original relative permeability of water phase without surfactant injection (dimensionless) k ow r The original relative permeability of oil without surfactant injection (dimensionless) K The average permeability (10 −3 μm 2 ) K i The initial permeability of injected solvent without considering the influence of gas (10 −3 μm 2 ) K mod The modified permeability of gas flooding (10 −3 μm 2 ) K pol The coefficient describing the influence of injected gas on viscosity (dimensionless) The coefficient of oil-phase relative permeability (dimensionless) M r The referenced mobility decreasing factor (dimensionless) M C7+ The molecular weight of heavy (C 7+ ) alkane MMP The minimum miscibility pressure of injected gas (MPa)

MMP impure
The minimum miscibility pressure of impure CO 2 -oil (MPa) MMP pure The minimum miscibility pressure of pure CO 2 -oil (MPa) MMP r The relative miscibility pressure (MPa) N o The geological reserves (t) P C2−6 The total molar percentage of C 2 -C 6 (dimensionless) Q Production rate (

Introduction
Currently, the main mathematical models for enhancing oil recovery (EOR) techniques simulation included compositional, black oil, streamline, and fractional flow theory models. The compositional model achieved reliable results and effectively simulated the seepage characteristics of injected solvent and crude oil. It required detailed physical data of rock and fluid, complex computation, and heavy workload (Li et al. 2019). Compared with the compositional model, the black oil model was simpler and required less data. The streamline model, a simulation method with improved computing speed, was derived from the black oil model or compositional model and presented result from the above model with lines perpendicular to equipotential line (Li et al. 2005;Wang et al. 2006). However, the accuracy of black oil model and streamline model still relied on considerable physical data of rock and fluid. The physical data of rock had included the compressibility, matrix of stiffness, porosity, permeability, and heterogeneity factor. The physical data of fluid had included the composition, solution, pressure, relative permeability, rheological property, adsorption quantity, interfacial tension, capillary force, gravity, and saturation. The fractional flow theory model was advantage of fast computing speed and low data demand by transforming three-dimensional flow to one-dimensional flow, but it simplified a certain parts of seepage characteristics (Kim 1990;Zhou 2017). Choosing a proper model for EOR technique evaluation would influence the result reliability as well as the period for EOR technique application. At present, a large number of oilfields are conducting research on the applicability and mechanism of different EOR techniques (Hosseini-Nasab and Simjoo 2018; Najimi et al. 2020;Khan et al. 2020). At the EOR evaluation stage of an oilfield, each oilfield comprises hundreds or thousands of different oil reservoirs, resulting in heavy work in EOR potential evaluation. Especially, in an oilfield developed with fluvial delta reservoirs, the complicated distribution of scattered small reservoirs in vertical and horizontal directions (Luo et al. 2021;Wang et al. 2021;Qu et al. 2021) brought trouble for evaluation work. It indicated the importance of a fast and reliable EOR evaluation model. Utilizing the conventional numerical model as well as introducing into many theoretical model or fitted equations, the accuracy of different EOR techniques was improved and mechanisms were considered. However, the numerical simulation for a specific reservoir could not be down before the concrete values or the similar scopes of physical data of rock and fluid were given. Many parameters in the theoretical seepage models of the EOR techniques were also hard to be estimated for a specific new layer. For example, Li et al. (2019) gave the empirical model of relative gas-phase permeability of air-foam flooding, whereas the unknown value of dimensionless interpolation parameter relied on another thirteen fitted values, and the mobility reduction factor MRF was obtained by the experiment (Hadian Nasr et al. 2020). The improved theoretical seepage model of EOR techniques was mainly used for simple physical model (Pu et al. 2015;Ahmed et al. 2020), on the other hand. Time-consuming on obtaining the unknown values and simulation with complex seepage model were unrealistic and uneconomical for the new layers evaluation. Thereby, improvement on the conventional seepage model and efficient numerical simulation method were essential for oil recovery and production effect estimating actually.
Focusing on fast evaluation on three EOR techniques (gas flooding, foam flooding and surfactant flooding), corresponding evaluation model was established in this paper by combining fractional flow theory with auxiliary equations. Modifications were made to the fractional flow rate in the auxiliary equation, according to the effect of heterogeneity and mechanism of different EOR technique. Finally, a comparison was made between the results from the modified fractional flow theory and those from conventional model by Eclipse software.

Mathematic model
The mathematic model for EOR technique applicability evaluation consisted of main equations based on fractional flow theory, equations for rapidly evaluation (describing the influence of factors on fractional flow in complex distributed porous media), and auxiliary equations for various EOR techniques.

Main equations based on fractional flow theory
The basic assumptions of the fractional flow theory model were as follows (Wang 2018): three phases (oil, gas and water) were considered and their flow followed Darcy's Law; the flooding was in an isothermal process and the viscous fingering was described with Koval coefficient; when adopting alternating injection method, the injected fluid was flowed into the porous media simultaneously at a fixed proportion; the porous media was without large cracks, free gas existed as well as leakage of injected gas; an irreversible adsorption happened and followed mass adsorption law; and diffusion and dispersion were considered.
The mass conservation equation of fractional flow theory was as below: where i = 1 represented water component; i = 2 represented crude oil component; i = 3 represented injected gas component; X D was the dimensionless distance, X D = X/L; t D was the dimensionless time for fluid passing by the hole, t D = ∫ t 0 qdt dV p ; F i was the total fractional flow of component i; C i was total concentration of component i.
The total concentration and fractional flow of several components can be expressed as: where C ij was the concentration of component i in phase j; j = 1 represented water phase; j = 2 represented oil phase; j = 3 represented injected gas phase; S j was saturation of phase j; f j was the fractional flow of phase j.
If injected gas and crude oil achieved miscible displacement, Eqs. (2) and (3) are simplified into a two-phase conservation equation (Burke 1984): When the term F i in Eq. (1) was divided by C i and the partial differential equation can be further expressed as: In Eq. (6), a concentration rate (fixed) was defined and expressed as From Eq. (7), two eigenvalues of were expressed as ± (Eq. 8). ± reflected two component lines, fast line and slow line, and they represented the different transfer direction of multi-phases. The fast line had to pass the initial conditions (in the oil reservoir), whereas the slow line had to pass the (1) injection conditions (Rogers and Grigg 2001;Knudsen and Hansen 2002).
The model of fractional flow theory firstly carried out two-phase flash calculation and fractional flow computation along two component lines. After the fast line and slow line were determined, their intersection point where the components transferred between the fast line and the slow line was determined ( Fig. 1). According to the fast and slow lines, the variations of concentration and fluid fractional flow within the affected range of gas flooding could be obtained. Combining with the mass conservation equation (Eq. 1) and calculation method of different seepage models, the oil output and oil reservoir recovery could be further calculated.

Equation for fluid output
(1) Fluid output before fluid breakthrough Before fluid breakthrough, the output of oil, water and injected solvent was calculated as below: where Q o , Q s and Q w were the output of oil, water and injected solvent, respectively; t i was the calculation time (time step i);F o,init was the total fractional flow of component oil at the initial time; Q was the production rate; B o was the volume factor of oil; F 3F t i was the total fractional flow of component 3 at calculation time t i ; Q r was the injection rate; B w was the volume factor of water.
(2) Fluid output after fluid breakthrough After the breakthrough, the output of oil, water and injected solvent was given below.
where F STL was the contribution of water drive on oil recovery in area without gas intrusion, F STL = E A ∕ T D2 ; E A was the area sweep coefficient; T D2 was the dimensionless time equivalence to the ratio of gas injection volume in local area to gas injection volume of whole well pattern; F 2F was the total fractional flow of component 2; t D1 was the dimensionless time equivalence to the ratio of (gas and water) total injection volume to the volume of intrusion area; B s was the volume factor of injected solvent.
(3) Oil recovery The oil recovery was calculated as below: where E R was oil recovery; Q o t N was the oil output at time step t N ; N o was the geological reserves.

Equations for influence of heterogeneity on fractional flow for rapidly evaluation
Expect for the main equation of fractional flow model, characters describing the complex distribution of petrophysics or rock physics in actual reservoir would influence the complex seepage. The factors were like such as viscous fingering, area sweep coefficient, vertical heterogeneity, and gravity (Tan and Homsy 1992). In the model for rapid evaluation of EOR techniques applicability, a simple mathematic description on the mechanism of factors influencing fractional flow would improve the solution efficiency.
Herein, Koval coefficient was used to describe the influence of heterogeneity on fractional flow. The fractional flow where K oval was the Koval coefficient; i (i = 1, 2, 3) was the viscosity of component i.
(1) Heterogeneity expressed by flow ratio The influence of viscous fingering and longitudinal heterogeneity on fractional flow could be considered by the Koval coefficient, and the expression was shown below.
where H gen was the coefficient related to heterogeneity; the coefficient component about inlayer heterogeneity ; D P was the value reflecting inlayer heterogeneity; V DPL was the value reflecting interlayer heterogeneity.
(2) Heterogeneity caused by gravity effect During gas flooding or foam flooding, because the gas density was less than the liquid density, the injected gas moved toward the top of the reservoir. The increased value of Koval coefficient was considered to express the influence of gravity separation.
w h e r e F ACT wa s t h e c o e f f i c i e n t r e f l e c ting the influence of gravity on gas flooding; the F ACT = 0.565 × lg D grav + 0.87 ; D grav was the dimensionless coefficient for gravity separation effect, D grav = 2.5271KK v A w − g ∕ Q rg g ; o and g were the initial viscosity of oil phase and gas phase, respectively; K was the average permeability; A was the well-pattern area; Q rg was the gas injection rate.

Auxiliary equations for seepage models of gas flooding, foam flooding and surfactant flooding technique
Mathematic equations describing the mechanism of different EOR techniques were talked in many references for the reliability and calculation efficiency. The mechanism was usually (12) 4 expressed in the main equation or auxiliary equations, and many styles of auxiliary equation were derived. Here, fitting equations or simple auxiliary equations of seepage model were adopted to describe the mechanism of different EOR techniques. The simple form of auxiliary equations brought efficiency for the rapid evaluation of EOR technique.

Relative permeability equation
Relative permeability equation was referred to that of BC model and is shown in Eq. (15) (Zheng et al. 2018). Chen (1999) as well as Ataie-Ashtiani and Raeesi-Ardekani (2010) listed several classic models, which were widely used in reservoir engineering.
where k rj and k ro were the relative permeability of j phase (j = water, gas, injected solvent) and oil phase, respectively; C rj , B rj , A rj , C ro , B ro and A ro were modification parameters to make the numerical relative permeability curve fitting with the actual relative permeability curve; S j was the saturation of j phase; S jr was the irreducible saturation of j phase; S orj was the residual oil saturation when coexistence of oil and j phase; 1 was the characteristic soil parameter, characterizing the pore-size distribution; m was the coefficient of oilphase relative permeability.

Gas flooding model
The EOR mechanism of gas flooding mainly included that: (1) reducing the viscosity of crude oil; (2) improving the oil-water mobility ratio; and (3) miscible drive. When the injected gas was dissolved in crude oil, the oil viscosity decreased. The injected gas that was both dissolved in crude oil and water also increased the viscosity of water and then the oil-water mobility ratio increased, thereby allowing oil phase to flow easily. With a certain composition of oil, certain temperature, and enough pressure, the injected gas and crude oil could reach miscible or near-miscible state.
The miscible state was with disappeared or ultra-low gas-oil interfacial tension to achieve efficient oil displacement (Jia et al. 2019). Considering the above EOR mechanism of gas flooding, modification was made for the fractional flow theory model. The viscosity of crude oil was modified by virial expansion method, and a minimum miscibility pressure calculation method was introduced to simulate the critical condition of miscible and immiscible displacement. The relative permeability equation was modified with miscible phase factor to simulate interfacial tension variation.
(1) Modified oil viscosity During gas injection, the crude oil viscosity was modified by applying Virial expansion method (Rivas et al. 1992). This method did not consider the influence of pressure on crude oil density.
where K pol was a coefficient describing the influence of injected gas on viscosity, with a default value of 11.855, dimensionless; G was the volume ratio of injected gas and crude oil; og and o were the crude oil viscosity after gas injection and the initial viscosity, respectively.
(2) Equation of minimum miscibility pressure The minimum miscibility pressure was the judgement criteria between miscible drive and immiscible drive in gas flooding. A prediction equation of minimum miscibility pressure was introduced into fractional flow theory to distinguish miscible and immiscible drive, referring to the empirical equations of Yuan et al. (2004) and Liao et al. (2014). The minimum miscibility pressure of pure CO 2 -oil was used as the basic reference, and a correlation coefficient as in Eqs. (17) and (18) was introduced to represent the minimum miscibility pressure of impure CO 2 -oil.
where MMP r was the relative miscibility pressure; MMP impure was the minimum miscibility pressure of impure CO 2 -oil, MPa; and MMP pure was the minimum miscibility pressure of pure CO 2 -oil, MPa; T cr was the relative critical temperature of impure CO 2 . The prediction equation of minimum miscibility pressure of pure CO 2 applied Yuan's method and shown in Eq. (19).
where i (i = 1, 2, …, 10) was the fitting coefficient; M C7+ was the molecular weight of heavy (C 7+ ) alkane; P C2−6 was the total molar percentage of C 2 -C 6 ; T was the reservoir temperature.
By applying the above prediction equation, the minimum miscibility pressure of collected crude oil was compared with results measured in the laboratory. As shown in Table 1, the new model's accuracy was improved significantly compared with other prediction models (Kovarik model, Sebastian model, Alston model, Yuan model) widely applied (Alston et al. 1985;Sebastian et al. 1985;Lansangan et al. 1993;Liao et al. 2014), and the error was maintained below 10%.  (3) Modification of relative permeability The interfacial effect of EOR mechanism during gas flooding could be achieved by modifying the relative permeability. This study introduced a miscibility factor (Eq. 20) to modify the relative permeability.
where is the defined miscibility factor; MMP was the minimum miscibility pressure of injected gas.
Through the miscibility factor, the permeability of injected solvent mixture (including gas) could be modified as follows: where K i was the initial permeability of injected solvent without considering the influence of gas; K mod was the modified permeability of gas flooding, equating 1 under complete miscibility state, and equating K i under complete non-miscibility.
An example of the modified relative permeability curve is shown in Fig. 2, and the miscibility degree (miscibility factor introduced) changed the permeability of injected solvent mixture as well as the relative permeability of oil phase. The endpoint value K rg (S o = 0) of relative permeability curve under partial miscibility state could be determined through interpolation calculation with Eq. (21). This endpoint value was used to recalculate the relative permeability under different oil saturation.

Foam flooding model
The main mechanism of foam flooding was to reduce the mobility ratio of foam (water) phase to oil phase. The foam-oil mobility ratio varied with different foam density, oil saturation and gas saturation. Based on the fractional flow theory and auxiliary equations of gas flooding model, the foam flooding model was got with additional necessary modifications (considering foam adsorption and reduced foam-oil mobility ratio).
(1) Foam adsorption The calculating equation of foam adsorption ratio is referred to that of US Department of Energy as Eq. (22) (Liao et al. 2014).
where DS was the adsorption ratio; was the rock porosity; rock was the rock density (g/ml); foam was the foam density (g/ml); A s was the proportion of clay mineral, dimensionless; and C foam was the volume concentration of foam, dimensionless.
The variables were suggested with default values in the rapid evaluation model, as the rock density being 2.68, the foam density being 1.0 and the proportion of clay mineral being 0.33.
(2) Mobility decreasing factor The mobility decreasing factor was introduced to simulate the mechanism of decreasing mobility and improving sweep factor in foam flooding. The mobility decreasing factor is expressed as Eq. (23) (Vargo et al. 2000;Hosseini-Nasab and Simjoo (2018).
where M r was the referenced mobility decreasing factor, dimensionless; F s , F w , F o , and F c were the mobility decreasing coefficient caused by foam concentration, water cut, oil phase, and flow velocity, respectively, that are dimensionless.
The calculating equations for the mobility decreasing coefficients were shown below. The mobility decreasing coefficient F c was related to flowing velocity and could be neglected in low permeability porous media due to ultra-low flow velocity. where C s was the effective foam concentration; C r s was the referenced foam concentration; and es was the control index; a was a coefficient relevant with water cut; f w was the weighting coefficient that controls the mobility ratio; S w was the water saturation; S l w was the minimum water saturation of foaming; S o was the oil saturation; S m o was the maximum oil saturation of foaming; eo was the transition index.

Surfactant flooding model
The EOR mechanism of surfactant flooding mainly included: (a) reducing the O-W interfacial tension and improving displacing efficiency; (b) crude oil on the surface of rock could be dispersed, stripped and form oil-in-water (O/W) emulsion so as to improve the mobility ratio of oil to water; (4) surfactant adsorption could change the wettability of the rock surface and make the rock surface shift from oil-wet to water wet or water-wet to strong water-wet (Yu et al. 2012). Focusing on interfacial tension reduction, surfactant adsorption and crude oil emulsification, the surfactant flooding model was got with additional necessary modifications on fractional flow theory.
(1) Surfactant adsorption The calculating equation of surfactant adsorption ratio referred to that of US Department of Energy as Eq. (15). The effective surfactant concentration was the product of injected surfactant concentration and surfactant adsorption ratio as Eq. (25).
where C surf and C surf,0 were the effective surfactant concentration and injected surfactant concentration, respectively.
The variables were suggested with default values in the rapid evaluation model, as the rock density being 2.68, the surfactant solvent density being 0.8 and the proportion of clay mineral being 0.33.
(2) Viscosity of surfactant solvent During surfactant injection, the water viscosity was influenced by the dissolved surfactant and decreased interfacial force. Resemble as Eq. (9), virial expansion method was applied to describe the viscosity of surfactant solvent (Elraies and Tan 2010;Vargo et al. 2000) and the influence of pressure on density was not considered as well. Here, the concentration of surfactant used the effective surfactant concentration. (24c) where surf was the effective viscosity of surfactant solvent; K pol was the viscosity coefficient, dimensionless, with a default value of 11.855; w was the initial viscosity of injected surfactant solvent.
(3) Interfacial tension The interfacial tension varied with surfactant concentration. Schechter model used in references was as follows (Burke 1984;Schechter et al. 1994), which described the relation between the minimum interfacial tension and the maximum interfacial tension. where os , min os , and max os were the instantaneous interfacial tension, minimum interfacial tension, and maximum interfacial tension between the oil and surfactant solvent, respectively; C OS , C min OS , and C max OS were the surfactant instantaneous concentration, the concentrations corresponding to the minimum interfacial tension and the maximum interfacial tension, respectively; es was the exponent parameter; a−b was the interfacial tension between two phases (a and b); Δ a and Δ b were differences of number density in two phases, Δ a = a∕A − a∕B , Δ b = b∕B − b∕A ; a∕A , a∕B , b∕B and b∕A were the number densities of molecules a and b in phases A and B, respectively; L a and L b stood for the number densities of pure molecules a and b in liquid, respectively; a and b were the surface tensions of pure molecules a and b, respectively; d aa and d bb were the hard sphere diameters of molecules a and b, respectively; ab was the cross diameter.
(4) Relative permeability The permeability and relative permeability both changed due to interfacial tension. A coefficient as Eq. (28) is introduced to describe the influence of interfacial tension on the relative permeability (Jones et al. 2016).
where F was the modified coefficient of relative permeability due to interfacial tension, which is dimensionless; os was an interfacial tension between oil and surfactant solvent phases; ref was the referenced interfacial tension, with a default value of 10; S oros and S or were the residual oil saturation after and before surfactant injection, respectively; k r was the oil-phase relative permeability after surfactant injection; k ow r and k m r were the original relative permeability of oil phase and water phase without surfactant injection, respectively.

Numerical simulation and discussion
As mentioned in above equations, several aspects were done to improve the calculating efficiency: one, using the fractional flow theory; two, simple equation for describing the factors influencing fractional flow in complex distributed reservoir; three, the variable in EOR models was suggested with default values in nearby area or with similar properties. The numerical calculation procedure of rapid evaluation model is shown in Fig. 3 (Wang et al. 2006).
The model was applied for a low permeability reservoir in Ordos basin, and the basic parameters are shown in Table 2. The rapid evaluation model and software Eclipse (Xiao et al. 2018;Haq et al. 2019) were used to predict the EOR potential of the oil reservoir. The numerical simulation result by gas flooding (2 year) with Eclipse is shown in Fig. 4. Because of the obvious heterogeneity of reservoir (distributed as permeability and porosity), the oil recovery at different place varied and gas drive fingering was found in the diagonal direction. A comparison on the predicted results from Eclipse and that from the rapid evaluation model is shown in Figs. 5, 6, 7.
The calculated results of the rapid evaluation model with fractional flow theory were close to that of Eclipse, with a difference within 10% on average, which implied good agreement of results. However, the computation speed of the rapid evaluation model was much faster than that of Eclipse. The computation time with rapid evaluation model (5 min) was shorter for 4.5 min than that with Eclipse (9.5 min) when numerical simulation for gas flooding and shorter for 8.7 min than that with Eclipse (15.3 min) when numerical simulation for foam flooding.  The main reason for the results' difference from two calculation methods was that Eclipse adopted the finite difference method and considered more factors (such as capillary effect). Besides, the numerical simulation model of Eclipse showed heterogeneity of oil reservoirs with inputting concrete rock properties, but the fractional flow theory model merely adopted heterogeneity character from numerical treatment method. The other factors and mechanisms of different EOR techniques were similarly considered. The rapid evaluation model was useful because much work needed to be done for evaluating amount of small reservoirs, whereas many basic parameters could not be quickly prepared or even found for oil recovery calculation. A comparison showed that the computational accuracy of the modified fractional flow theory model could satisfy preliminary evaluation on large-scale EOR technique. Moreover, the model could further provide technical support for EOR technology optimization of an entire oil field. A further numerical simulation on the effect of injection volume on oil recovery of different EOR techniques had indicated that the foam flooding had a large oil recovery, whereas the gas flooding had a low oil recovery. The difference in oil recovery of various EOR technique depended on the reservoir properties. The strong permeability heterogeneity (existence of cracks and ultra-low permeability matrix) would easily cause obvious channeling, especially gas channeling with high mobility. The rapid evaluation model was efficient for predicting the optimal injection volumes ( Fig. 8) of different EOR techniques. The optimal injection volume was about 0.35-0.65 pore volume of foam flooding, about 0.5-0.8 pore volume of surfactant flooding and about 0.6-1.0 pore volume of gas flooding, respectively, for the numerical area and parameters in Table 2.

Conclusions
• The rapid evaluation model was proposed for EOR technique evaluation and based on the modified fractional flow theory, which transformed the seepage to one-dimensional flow. The model considered the equation for fluid output before or after fluid breakthrough, the influence of heterogeneity on fractional flow, fitting equations or simple auxiliary equations, and the EOR mechanism of gas flooding, foam flooding and surfactant flooding. • A comparison on the predicted results from Eclipse and that from the rapid evaluation model had shown that the rapid evaluation model was feasible and efficient to evaluate the proper EOR technique. The computation time of numerical simulation for gas flooding and foam flooding with rapid evaluation model was shorter for 4.5 min and 8.7 min, respectively. More computation time was saved during gas flooding simulation, and less time was saved during foam flooding simulation.
• A rapid evaluation model for EOR techniques applicability was beneficial for improving numerical efficiency and optimizing injection parameter. The optimal injection volume was about 0.35-0.65 pore volume of foam flooding, about 0.5-0.8 pore volume of surfactant flooding and about 0.6-1.0 pore volume of gas flooding, respectively, based on the numerical area and parameters.
Funding The project is funded by Natural Science Basic Research Plan in Shaanxi Province of China (2020JM-299) and Open Foundation of ShaanxiKey Laboratory of Carbon Dioxide Sequestration and Enhanced Oil Recovery.

Conflict of interest
The authors declare that we have no conflicts of interest.
Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors.
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