Modified cell-to-cell method for slim tube test simulation considering the porous media effect

Abstract

This study estimates the minimum miscibility pressure (MMP) for a hydrocarbon gas injection project using integrated laboratory and simulation approaches. The MMP was first determined experimentally via a slim tube displacement test. Concurrently, a modified cell-to-cell simulation (CCS) model was developed, incorporating key corrections to account for porous media effects. These modifications included: adjusting the Peng–Robinson equation of state for a porous medium, improving the critical properties shift equation, incorporating capillary pressure into vapor–liquid equilibrium calculations, and refining fluid transfer criteria. The simulation used the permeability-to-porosity ratio to represent the porous medium’s influence on phase behavior. Results demonstrate that a decreasing permeability-to-porosity ratio reduces the MMP. This effect is particularly significant in tight reservoirs (with ratios below 10), indicating that miscibility can be achieved more readily in such formations compared to conventional reservoirs. The modified CCS method provides an efficient and reliable tool for MMP estimation.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-38525-4.

Introduction

Nowadays, due to the decline in the natural production potential of oil reservoirs, the need for effective enhanced oil recovery (EOR) operations is more pressing than ever. Various methods are being used to increase extraction from oil reservoirs, both experimentally and operationally. The most important EOR methods are gas injection^1–3, chemical injection, and thermal methods. Among the abovementioned methods, gas injection is one of the most efficient EOR methods, especially in heavy oil reservoirs. In gas injections, as an EOR method (not as pressure maintenance), which is usually performed in a miscible, gas injection is attempted in conditions where oil and gas completely dissolved in each other. In this case, the key parameter in gas injection operation is the minimum miscibility pressure or MMP⁴.

The success of a gas injection project hinges on achieving miscibility between the injected gas and the reservoir oil, a condition met at or above the minimum miscibility pressure (MMP). The MMP is defined as the lowest pressure at which, through a multi-contact process, the two phases become dynamically miscible (their compositions converge, and interfacial tension (IFT) vanishes to effectively zero^5–9. Mechanistically, this occurs because sufficient pressure enables the intense, reciprocal mass transfer (vaporization and condensation) of intermediate hydrocarbon components needed to eliminate the phase boundary¹⁰. Consequently, operating above the MMP is critical as it shifts the displacement mechanism from immiscible (governed by unfavorable phase behavior and capillary trapping) to miscible, leading to a dramatic increase in microscopic displacement efficiency and ultimate oil recovery¹⁶.

MMP can be determined using several laboratory and computational methods^6,11. Among the laboratory methods introduced to measure MMP, the slim tube displacement is the most reliable method because it is performed dynamically. More importantly, the miscibility process occurs in a porous medium, i.e., the porous media effect is considered. On the other hand, the MMP is determined based on the ultimate recovery factor variation due to miscibility¹².

In addition to laboratory methods of MMP measurement, there are various computational methods for calculating and estimating the MMP value. Due to the high cost and time-consuming nature of laboratory methods, along with the need to determine MMP under different conditions, computational methods can be valuable for investigating different conditions and analyzing the sensitivity of parameters¹³. Among the most common methods for estimating and calculating the MMP are empirical correlations and approaches based on equations of states (EOSs). Empirical correlations, which are also obtained based on laboratory data, can calculate the MMP under the specific conditions of each correlation. Therefore, a method based on EOS is preferred for MMP approximation. The most important methods based on EOS are cell-to-cell simulation (CCS)^14–16, method of characteristics (MOC)¹⁷, and one-dimensional fluid flow^18,19. In the abovementioned methods, the phase behavior of the fluids should be accurately simulated using the tuned EOS. It should be noted that in recent years, the use of various methods of artificial intelligence (AI)^20–22 and molecular dynamics (MD) simulation²³ to estimate the MMP has attracted the attention of researchers.

A recent issue with EOS-based methods for MMP estimation is accurately capturing phase behavior in porous media. Fluid phase behavior can be studied under two conditions: bulk fluid and confined fluid²⁴. In a bulk fluid, a molecule is surrounded by other fluid molecules, whereas the molecules of a confined fluid are surrounded by other fluid molecules and the solid medium in which the fluid is placed²⁵. In other words, when fluids are placed in small pores, the mean free path of the fluid molecules is reduced, and the fluid-wall interaction plays a vital role in fluid behavior²⁶. In the porous media, fluid molecules, especially those of heavy components, are adsorbed onto the wall of the porous media, which changes the component’s critical properties²⁷. Considering the size of the pores and throats of the porous medium, the thickness of the adsorption layer cannot be ignored compared with the pore diameter²⁸. For the first time in 1986, Pederson et al.²⁹, by studying the density profile of Argon with changing pore radii and under different conditions using mean field theory, apperceived that the transition from gas to liquid phase arises at pressures lower than the vapor pressure of the bulk fluid. Zarragoicoechea et al.³⁰ (2004) showed through theoretical and modeling studies, as well as experimental data on single components, that the phase behavior of bulk-free fluid differs from that of confined fluid. They showed that changing the critical properties of fluid components concerning the molecule-to-pore size ratio can modify the behavior of confined fluids. The discrepancy between simulations using EOS and experimental results was first studied by Singh et al.³¹ in 2009. They performed Monte Carlo simulations to investigate the fluid confinement effect on vapor–liquid interfacial tension for some hydrocarbon components (CH₄, C₂H₆, C₃H₈, n-C₄H₁₀, C₈H₁₈) in graphite pores with mica surfaces. The results show that the critical temperature shift of the components mentioned depends on the nature of the surfaces. Additionally, changes in critical pressure are influenced by the pore size, increasing as the pore size decreases. In 2010, Travalloni et al.³² developed the van der Waals EOS to account for the fluid confinement effect on fluid properties. Cui et al.³³ (2018) modified the Peng–Robinson EOS using the theory of adsorption of heavy components of hydrocarbon fluids on the pore wall. They also combined capillary pressure with phase equilibrium and included its effect in the calculations. In 2019, Tan et al.³⁴ measured the critical points of carbon dioxide and ethane for the first time using the scanning calorimetry method and observed that the critical pressure and temperature of both fluids increased as the pore size decreased.

In the studies performed by Song et al.²⁸, a porous medium was modeled as a cylindrical tube. In this pore with a radius of, the thickness of the absorption layer is (Fig. 1).

Fig. 1.

General schematic of the adsorption model inside a cylindrical cavity where heavy molecules are adsorbed inside its wall.

According to Zhang et al.³⁵, the absorption zone thickness as a function of the pore radii and Lennard-Jones molecular diameter can be calculated using the following empirical equation:

1

In Eq. (1), is the pore radius, is the Lennard-Jones molecular diameter, m and n are the constant parameters. The Lennard-Jones molecular diameter can be calculated as follows:

2

In Eq. (2), and represent the critical pressure and temperature of a hydrocarbon component in bulk fluid condition, respectively. The two constant parameters m and n are also calculated as follows:

3

4

where is the molecular weight of each hydrocarbon component.

Song et al.²⁸ provided an empirical equation to calculate the reduced absorption density in terms of the Lennard-Jones molecular diameter, pore diameter, and thickness of the absorption region, as follows:

5

In Eq. (5), is the Lennard-Jones diameter. Therefore, they modified the Peng-Robinson EOS for a cylindrical porous medium as follows:

6

where, is defined as

7

In Eq. (6), P is the pressure, T is the temperature,is the molar volume, R is the global gas constant, a and b are the constant of Peng-Robinson EOS. Calculating the fluid properties using the modified Peng-Robinson EOS for a single cylindrical pore²⁸, results in values that differ from those of the bulk fluid.

Parallel to the studies of the phase behavior of confined fluids, slim tube simulations have been conducted using cell-to-cell methods. For the first time, Metcalfe et al.¹⁶ in 1972, using a multi-cell method, simulated the process of gas injection into a petroleum system to estimate the MMP. The proposed method involves placing several cells one after the other, and at the beginning of the simulation, all the cells are saturated with oil in the reservoir condition. A certain amount of gas is then added to the first cell, where it is assumed to have reached a complete thermodynamic equilibrium with the primary oil in that cell. The excess volume, relative to the cell volume, is transferred to the next cell, accounting for transfer mechanisms such as stagnant oil, excess oil, and the mobility of phases. In the second cell, the new composition is determined by considering the added fluid, and the vapor–liquid equilibrium calculations are performed again for this cell. This calculation is performed for all cells until the last cell. Finally, the fluid output from the last cell is considered the production fluid from the slim tube. In 2006, Zhao et al.³⁶ modified the cell-to-cell algorithm by proposing a computational method for measuring the mobility transfer of phases. In other words, according to their suggestion, the movement of one phase from one cell to the next cell is performed by considering its relative permeability and viscosity. In 2018, Zhang et al.³⁷ simulated the slim tube experiment by capillary pressure consideration in the calculation of phase behavior and using cell-to-cell simulation. In 2021, Sun and Li¹⁴ conducted cell-to-cell simulations by capillary pressure consideration in equilibrium calculations as well as the critical properties shift of confined fluid components (using the correlations by Tan et al.³⁴. They also simultaneously used the two criteria of oil recovery and tie line length for MMP estimation. The study follows the same algorithm presented by Jabert et al.¹⁵, modified for a single pore as a porous medium (confined).

To data, many studies have been conducted to calculate the critical properties of porous media and to modify cell-to-cell simulation (CCS) for slim tube modeling. Finally, according to the best method presented by Sun and Li¹⁴, the porous media is presumed to be a single tube, and calculations have been performed for it. If in reality, it is necessary to consider the real properties of a porous media such as porosity and permeability in simulations. This paper aims to modify the cell-to-cell simulation algorithm presented by previous researchers to better represent real porous media. Therefore, the critical properties shift in based on the porosity and permeability of the porous media, the modification of Peng-Robinson EOS based on the porosity and permeability of the porous medium, the effect of capillary pressure on equilibrium calculations, and the phenomenon of gas channeling in the modified cell-to-cell simulation (MCCS) are most important objectives and novelties of this investigation.

According to the objectives of this research, the second section discusses methodology, including experiments and modeling methods. In the third part, the results of this research are presented and discussed in detail. Finally, in the fourth section of this paper, the most important deduced conclusions and suggestions for future work are stated.

Methodology

Experiments

To carry the slim tube test out, live oil was provided from a reservoir in southwest Iran using the downhole sampling method. The bubble-point pressure and gas-oil ratio (GOR) of this live oil at 100 °C were measured as 17 MPa and 87.27, respectively. In addition, the composition of the live oil was determined using gas chromatography (GC) analysis (Hydrocarbon compositional analyzer, VINCI, France), as presented in Table 1. In addition, to obtain laboratory data for EOS tuning in modeling, constant compositional expansion (CCE) and differential liberation (DL) tests were performed on oil samples (FluidEVAL, VINCI, France). The injection gas in this study is also considered a hydrocarbon gas sample, whose composition (Determined by GC analysis) is shown in Table 1.

Table 1.

Composition of the reservoir oil and injected gas.

Components	Live oil (mole %)	Injected gas (mole %)
CO2	0.1504	1.24
N2	0.1273	5.89
C1	33.8994	45.92
C2	8.4226	22.19
C3	8.691	11.06
iC4	1.9773	8.75
nC4	2.8119	1.59
iC5	0.9752	0.46
nC5	1.158	0.32
C6	1.7783	1.15
C7	2.1529	1.42
C8	4.3934	0.0
C9	2.2126	0.0
C10	1.5804	0.0
C11	1.041	0.0
C12+	28.6284	0.0

C12 + properties:.

Molecular Weight: 318 kg/kgmol.

Density: 948 kg/m³.

To carry out the slim tube test, the slim tube is first completely saturated using industrial toluene at the desired test pressure. Then the live oil is injected into the slim tube, resulting in toluene production.

The outlet of the slim tube is monitored using a sight glass and a camera. During the injection of oil into the tube, because of the removal of toluene from it, the fluid inside the sight glass has a bright color. Oil injections continue until the color of the fluid inside the sight glass darkens. This indicates that the toluene inside the slim tube has been removed, and the injected oil has reached the output. In this case, when the slim tube was completely saturated with oil at the pressure of the test, 24 h was given to the apparatus (STS700, VINCI, France) to reach the temperature of the test (100 °C). Finally, the desired gas is injected into the slim tube at the pressure and temperature of the experiment and a constant flow rate (0.08 cc/min) by a gas injection pump. Gas injection continues up to 1.2 pore volume of the slim tube. During gas injection, the oil produced from the slim tube is collected by a burette and recorded over time using a camera. After gas injection is complete, the slim tube is cleaned by re-injection of toluene and then prepared for testing at the next pressure. The entire process is repeated under different test pressures. Usually, the slim tube test is performed at five different pressure points. Figure 2 shows the slim tube apparatus used in this study. The specifications of the slim tube are presented in Table 2.

Fig. 2.

Slim tube apparatus used in this study.

Table 2.

Slim tube apparatus specifications.

Properties	Values or specifications
ID of the slim tube (mm)	4.00
Length of the slim tube (m)	12.20
Slim-tube material	316 stainless steels
Pore volume (cc)	100.6009
Permeability (D)	3.6
Porosity (%)	29.0

Modeling

Modification of Peng-Robinson EOS for porous media

In this study, considering the primary characteristics of a porous medium, i.e., porosity and permeability, it is necessary to generalize the Peng-Robinson EOS modified by Song et al.²⁸ to a natural porous medium. The Peng-Robinson EOS has been explained in Supplementary Appendix A.

The equations for a single cylindrical pore are presented in the model presented by Song et al.²⁷. However, in a porous medium, it is necessary to relate a single pore to the properties of the porous medium. The most important porous medium properties are porosity and permeability. Therefore, instead of considering a single pore, it is necessary to include these two characteristics of the porous medium in the equation presented for the thickness of the absorption zone. One of the most important assumptions of this study is the use of a simplified model of a porous medium called bundle of tubes. In fact, given that the change in critical properties has been measured and modeled in pipes of different sizes³⁴, the bundle of tubes model seems appropriate for modeling porous media. However, it should be noted that the porous medium in a slim tube consists of grains with spherical geometry, which is different from bundle of tubes model. This discrepancy can cause differences in the final results of the slim tube test and its modeling. For this purpose, a model for the porous medium is considered, in which instead of a single cylindrical pore, many cylindrical pores are located next to each other, as shown in Fig. 3. According to this figure, several parallel capillary tubes are placed next to each other. The flow equation in the n capillary tube can be expressed (according to Poiseuille’s equation³⁸) mathematically using Eq. 8.

8

Fig. 3.

Model for the hypothetical porous medium.

where, q is flow rate, is fluid viscosity, is the pressure difference between the two sides of the tube, L is the tube length, and is the tube radius. On the other hand, the total flow rate from the hypothetical porous medium can be expressed as follows, considering the permeability parameter and relying on Darcy’s law (Eq. (9)).

9

where K is the average absolute permeability and A is the total cross-sectional area of the porous medium. Therefore, by equating Eq. (8) and Eq. (9), the permeability of the hypothetical porous media can be determined as follows:

10

On the other hand, the porosity of the porous media can be defined as the ratio of the pore volume to the total volume (Eq. (11)).

11

According to Eq. (11), the value of the cross-sectional area of the porous media shown in Fig. 3 can be written as follows:

12

Therefore, by substituting the Eq. (12) in Eq. (10) and solving it according to the radius of the pores, gives:

13

In Eq. (13), the pores radius of the porous medium can be calculated in terms of average permeability and porosity. Therefore, instead of using the pore radius in the calculations, the expression presented in Eq. (13) can be used, which gives an average radius in terms of the average permeability and porosity of a hypothetical porous medium.

Therefore, if Eq. (13) is substituted in Eq. (1) the thickness of the absorption zone is obtained as a function of the porosity and permeability of the porous medium as follows:

14

Additionally, by substituting Eq. (13) into Eq. (5), the equation of the reduced absorption density of the porous medium is obtained as follows:

15

By substituting Eq. (14) into Eq. (15) in the suggested Peng-Robinson EOS by Song et al.²⁸ for the single cylindrical tube, the modified Peng-Robinson EOS for a real porous medium will be obtained:

16

An inflection point occurs at the critical point, for all cubic equations of state. In mathematical terms, the first and second derivatives of pressure in terms of volume at a constant temperature at the critical point are equal to zero. Hence, we have:

17

which by solving Eqs. (16) and (17) for parameters a and b we will have:

18

19

where,and are the critical pressure and temperature, respectively, in the pore. By combining Eqs. (18) and (19), it is possible to calculate the critical pressure and temperature in the porous medium as follows:

20

21

Therefore, using the Eqs. (20) and (21), it is possible to calculate the critical pressure and temperature for a porous medium with specific permeability and porosity. To better compare the changes in pressure and critical temperature, we describe the changes in these two critical characteristics compared to the free bulk fluid state as follows:

22

23

where,and are the critical pressure and temperature, respectively, of the bulk fluid. To survey the effect of the porous media on the critical properties of the pure components, the changes in critical pressure and critical temperature in terms of the permeability to porosity ratio, according to Eqs. (22) and (23), are shown in Figs. 4 and 5, respectively. According to these figures, it can be seen that for all components, as the permeability to porosity ratio of the porous media decreases, the critical properties change, with these changes increasing accordingly. fluid molecules experience increasingly significant interactions with the pore walls. This enhanced solid-fluid interaction alters the effective intermolecular forces within the fluid itself, leading to the measurable shifts in critical properties. Furthermore, it is observed that as the fluid components become heavier in a fixed permeability-to-porosity ratio, the deviation of the critical properties from the state of the free bulk fluid increases. Heavier hydrocarbon components exhibit greater deviations from bulk critical properties at a fixed k/φ ratio. This is mechanistically explained by their larger molecular size and polarizability, which amplify their response to confinement. The increased surface area of heavier molecules leads to stronger van der Waals interactions with the pore walls, making their phase behavior more sensitive to the porous environment than that of lighter, smaller molecules.

Fig. 4.

Critical pressure changes with the permeability-to-porosity ratio for certain hydrocarbon components.

Fig. 5.

Critical temperature changes with the permeability-to-porosity ratio for certain hydrocarbon components.

Vapor–liquid equilibrium (VLE) calculations

To perform VLE calculations, it is necessary to check the stability of the phase first, which is conducted using the stability analysis algorithm provided by Michaelsen (1982)³⁹. If a two-phase system is detected, VLE calculations will be performed for the desired system. To perform equilibrium calculations, it is first necessary to solve the Rushford-Rice equation to calculate the mole fraction of the gas phase⁴⁰:

24

In Eq. (24), is the mole fraction of the gas phase and is the equilibrium ratio for different components of the fluid, based on the values obtained from the stability analysis, and is the overall composition of the fluid. Equation (24) will be solved for b using the Newton–Raphson numerical method and based on the initial values of the equilibrium ratio. Finally, the equilibrium ratio is calculated as follows:

25

Where,andare the fugacity coefficients of the ith component in the oil and gas phase, respectively, and are the oil phase pressure and capillary pressure in the porous media, respectively. Derivation of Eq. (25) and the fugacity calculation in porous media is presented in Supplementary Appendix B. According to Nghiem and Heidemann⁴¹, taking the logarithm of both sides of Eq. (25), improves stability in the calculations, as follows:

26

If Eq. (26) is written for each component, a system of nonlinear equations is formed, which can be solved using the quasi-Newton successive substitution (QNSS) method⁴¹. However, an important aspect of solving Eq. (26) is the need to determine the capillary pressure (in term of MPa) of the porous media. The capillary equation of a single capillary tube is:

27

Where, is equilibrium interfacial tension (IFT), is the contact angle, and r is the pore radius. In this study, due to the existence of a two-phase oil-gas system, the contact angle is assumed to be zero. Also, by substituting Eq. (13) into the Eq. (27) and calculating the unit conversion factor in the SI system, Eq. (28) is obtained:

28

where, is the equilibrium interfacial tension (IFT) in mN/m, and K are the porosity and permeability of the porous media, respectively. According to Eq. (28), it can be seen that the capillary pressure is a function of IFT, which is also a function of the results of the VLE calculations. Therefore, in this case, a series of iterative calculations will be performed to perform VLE calculations. Note that the equilibrium IFT is calculated from the Parachor model⁴². The desired iterative calculation algorithm is shown in Fig. 6.

Fig. 6.

General algorithm for performing Vapor–Liquid-Equilibrium (VLE) calculations.

Modified cell-to-cell simulation (MCCS) algorithm

According to the explanations presented in the first section about the cell-by-cell method for simulating the slim tube test, successive studies have been conducted to improve the calculations of this method and bring its results closer to the slim tube test results by previous researchers^{14–16,36,37,43–47}. According to the performed studies, modifications include considering the amount of absorption of molecules on the wall of the porous media, critical properties shift of fluid components due to being in a confined environment, introducing the capillary pressure effect in the VLE calculations, and using the mobility ratio of the injected gas and oil to determine how fluids are transferred from one cell to the next cell.

In this study, three additional modifications were made to the model previously presented. The first is the modification of the PR-EOS for a real porous medium by knowing the porosity and permeability (this modification has been explained in detail in the section “Modification of Peng–Robinson EOS for porous media”).

The second modification of the confinement effect is the phenomenon of gas channeling after the gas breakthrough. In other words, the two-phase mobility ratio criterion was applied before gas breakthrough. This involved estimating the mobility of both phases based on relative permeability and fluid viscosity, to determine the flow fraction of each phase. However, after the breakthrough, gas and oil no longer move in the porous medium according to their mobility ratio, but the gas that has found its way out of the porous medium continues to move. In other words, it attempts to advance from the oil phase with channelization. To simulate this phenomenon, after the breakthrough, the criterion of fluid transfer from one cell to the next cell is changed, and the moving excess criterion⁴⁸ is used to transfer fluids between cells. In this criterion, the gas phase is transferred to the next cell according to the amount of excess volume calculated, and if excess volume remains, the remaining volume is transferred from the oil phase to the next cell. In this case, the movement of the oil phase is due to oil swelling.

The third modification is to apply the effect of the capillary pressure in the VLE calculations, according to the section “Vapor–liquid equilibrium (VLE) calculations”. According to the abovementioned cases, the algorithm written in this research to simulate the slim tube using the cell-to-cell method will be as follows:

Step 1

First, the basic information of the slim tube test is determined to perform the simulation. These basic characteristics include pressure, temperature, gas injection flow rate, the composition of oil and injected gas, slim tube pore volume, number of simulation cells, as well as the porosity and permeability of the slim tube.

Step 2

According to the characteristics of the porous media, the modified critical pressure and temperature parameters are calculated using Eq. (22) and Eq. (23), respectively.

Step 3

At the test temperature and pressure, the isothermal compressibility (Z-factor) value is calculated for oil and gas.

Step 4

According to the injection gas flow rate used in the slim tube test and Jabert et al.¹⁵ suggestion to inject a volume equal to 33% of the volume of each cell into the first cell, the simulation time is calculated as follows.

29

Where, is the slim tube pore volume, is the injection gas flow rate, and is the total number of cells in the slim tube model. Then, according to the calculated time interval and the injected gas flow rate, the injected gas moles in each time interval to the first cell are calculated as follows:

30

where, P is the pressure, R is the global gas constant, T is the temperature,is the compressibility factor of the injected gas, andis the time step between two injection into the cell-to-cell model. In addition, the amount of the initial mole of in situ oil in the slim tube was calculated under the following test conditions:

31

Therefore, the total composition of fluids in the first cell after gas injection can be calculated as follows:

32

where, and are the total mole of oil and gas, respectively in the first cell at time t, and are the total composition of oil and gas, respectively in the first cell,is the injected-gas mole amount at each time step, and is the composition of the injected gas.

In the Eq. (32), and are the mole fractions of the ith component in in situ oil and gas, respectively, and is the mole fraction of the ith component in the injected gas. The Superscript t also shows the ongoing time step. In addition, for the second to last cells, the composition of the total fluid in the cell can be calculated from the following equation:

33

Where, and are the transferred oil and gas, respectively, from the previous cell, and superscript n and show the current and previous cell numbers, respectively.

Step 5

After determining the composition of the fluid in each cell, VLE calculations are performed (using the introduced algorithm in the section “Vapor–Liquid Equilibrium (VLE) calculations”) at the pressure and temperature of the experiment and considering the capillary pressure effect in each cell. The results of these equilibrium calculations are the oil composition, gas composition, and gas phase mole fraction in each cell. After VLE calculations in the first cell, the equilibrium oil and gas volume in each cell is calculated from the following relationships.

34

35

Where, is the gas mole fraction in the first cell. In the other cells, the equilibrium oil and gas volumes can be calculated using an equation similar to Eq. (34) and Eq. (35), but it should be noted that gas injection is performed only in the first cell.

Step 6

The saturation of each phase is calculated according to the equilibrium volume of each phase. Using the oil and gas saturation in each phase and the relations provided by Zhao et al.³⁶, the relative permeability value for the two phases is calculated as follows.

36

37

where,and are the gas in oil-residual saturation, respectively, in each cell. The viscosity value of each phase was also calculated using the Lohrenz-Bray-Clark (LBC) correlation⁴⁹. With the relative permeability and viscosity of the two phases being known, the gas -phase flow fraction can be calculated as follows:

38

where, and are the oil and gas viscosity, respectively.

Step 6

According to the assumption that the volume of the hypothetical cells considered in the cell-by-cell simulation is constant, the excess volume after performing VLE calculations, as well as the volume of oil and gas that must be transferred to the next cell, is calculated as follows.

39

40

41

where, and are the oil and gas volume, respectively in each cell,is the excess volume in comparison with the pore volume of each cell, and are the transferred oil and gas volume, respectively from each cell.

Step 7

Before the gas breakthrough phenomenon, the calculations in step 6 are used to calculate the oil and gas transferred volume. However, to consider the effect of gas channeling in the porous medium after the breakthrough, the volume of oil and gas transferred to the next cell can be calculated as.

42

The presented algorithm is performed until 1.2 of the slim tube pore volume is injected into the first cell. Another point in the cell-to-cell simulation is the effect of cell number on the final oil recovery. According to Jabert et al.¹⁵ suggestion, the simulation was performed in different numbers of cells and to eliminate the effect of dispersion in the calculations, the ultimate oil recovery was estimated at an infinite cell number.

In this section, the modified cell-to-cell simulation (MCCS) algorithm for slim tube modeling is explained in detail. Finally, the modified items in this model compared to previous models can be summarized as follows:

• A)Modification of Robinson-Peng EOS and equations for calculating the critical properties shift based on the permeability to porosity ratio of real porous media,
• B)Considering the mechanism of the gas channeling phenomenon in the porous medium after the gas breakthrough,
• C)Considering capillary pressure in the volume calculation for the oil and gas phases in each cell.

Results and discussion

The slim tube test using the desired hydrocarbon injection gas was performed at five pressure points: 20.00, 22.12, 24.13, 28.96, and 31.0 MPa at a temperature of 100℃. It is usually recommended to use three pressure points below the MMP and two pressure points above the MMP. It is also recommended that the selected pressure points are not located in areas close to the MMP, because in these areas, the graph of oil recovery versus pressure of the test has a curve for analyzing the data according to the criterion of the linear intersection (LI)^50,51, and it is not possible to use the points obtained in the pressure range close to the MMP. Furthermore, the injection flow rate was set to 4.8 cc/hr., and the injected total volume was equal to 1.2 PV (pore volume). At each pressure step, the oil recovery factor (ORF) is recorded as a function of the gas injection volume (Fig. 7). Finally, the graph of the ultimate oil recovery factor is drawn according to the test pressure in Cartesian coordinates. The results of the slim tube test are shown in Fig. 8. According to this figure, using the criterion of linear intersection (LI), the value of the MMP was estimated to be 25.28 MPa. There are different opinions on the use of various criteria for MMP approximation. According to the study of Zhang and Gu⁵⁰, there are two methods and general approaches to the criterion of linear intersection. The first approach is to find the intersection point of two fitted lines on the data of the immiscible and miscible parts. However, there is another approach after the condition of the slim tube reaches miscibility, and with the increase in pressure, the ORF remains constant. Therefore, MMP is obtained from the intersection of the straight line in the immiscible condition with the straight line with about zero slope in the last pressure step, which equals 25.28 MPa. According to Zhang and Gu⁵⁰, determining a single point for the MMP will have very low accuracy because of the nature of the slim tube test. They suggested that instead of providing a single point for the MMP, it should be replaced by an interval. In other words, an interval can be determined where the MMP is located. In this way, the uncertainty about the exact value of the MMP is reduced.

Fig. 7.

Oil recovery factor versus pore volume of gas injection for each pressure step.

Fig. 8.

Ultimate oil recovery versus pressure during the experiment, with MMP determined using the linear intersection (LI) criterion.

In the modeling of the slim tube test by modified cell-to-cell simulation (MCCS), calculations were performed according to the algorithm presented in the section “Modified cell-to-cell simulation (MCCS) algorithm”. Based on the studies of Jabert et al.¹⁵, increasing the number of cells leads to higher oil recovery at constant pressure. This occurs because the dispersion effect decreases with an increase in the cell number, until considering the oil recovery corresponding to the infinite cell number (), the dispersion effect reaches zero. In this simulation, which was performed at pressures equivalent to the slim tube steps, modeling was performed for the number of cells 20, 25, 50, 70, 100, and 200. The ORF is then drawn in Cartesian coordinates (Fig. 9). According to this figure, it can be seen that the ORF has a linear relationship with. If the cell number tends to infinity, the calculated value of this number of cells is considered to be the ultimate oil recovery factor for each pressure step.

Fig. 9.

ORF in terms of the inverse of the square of the number of cells in the cell-to-cell simulation.

If the ORF with an infinite number of simulation cells is plotted against pressure in Cartesian coordinates, the results shown in.

Figure 10 will be obtained. For better comparison, the simulation results and the slim tube test results are presented on the same graph. According to this figure, the simulation results are in good agreement with the laboratory results, and the value of the MMP obtained from the simulation using the LI criterion is estimated to be 26.11 MPa, which is only 0.83 MPa different from the MMP obtained from the slim tube test. In general, this estimate has a relative error of 3.2%. In addition, the trend of the graph obtained from the simulation is completely consistent with that of the slim tube test data. However, the modified CCS model demonstrates good predictive accuracy with a Mean Absolute Percentage Error (MAPE) of 5.52% across all test cases. The mean percentage error of + 5.52% suggests a slight overestimation tendency, though the 95% confidence interval (−0.73% to + 11.77%) indicates this trend requires verification with additional data. Overall, the model provides reliable MMP estimates suitable for screening and preliminary design, with predictions typically within approximately ± 5.52% of experimental values.

Fig. 10.

The results of the modified cell-to-cell simulation along with the results of the slim tube test.

Although the proposed model attempts to make the experimental and modeling results as close to each other as possible, there is still a discrepancy between them. This discrepancy could be due to the limitations of the proposed model. The first limitation of the model relates to the use of the bundle of tubes model, rather than a real porous medium. As previously mentioned, the porous medium inside the slim tube consists of glass beads of different sizes, while the bundle of tubes model consists of several parallel capillary tubes of the same size. While the bundle of tubes conceptual model provides a valuable and computationally efficient framework for incorporating porous media effects into phase behavior and MMP calculations, it is recognized that it simplifies the true complexity of natural pore systems. Real carbonate and clastic reservoirs exhibit heterogeneous pore geometries, including pores, throats, tortuous pathways, and mixed wettability. These features could influence the MMP prediction in several ways. First, enhanced capillary pressure effects in smaller, irregular throats might amplify the shift in critical properties and phase equilibrium beyond what is predicted by a uniform permeability-to-porosity ratio. Second, increased tortuosity and dead-end pores could hinder efficient mixing and compositional exchange between oil and gas, potentially requiring higher pressures for full miscibility to develop dynamically at the core scale, even if the thermodynamic MMP at the pore level is reduced. The proposed model, focused on establishing a fundamental thermodynamic trend, implicitly averages these geometric effects into the permeability and porosity terms.

However, the wettability of the porous medium is another critical factor influencing interfacial phenomena and phase behavior in real reservoirs. Recent advances in molecular dynamics (MD) simulations, such as those reviewed by Jia et al. (2025), provide a fundamental understanding of how nanoscale wettability, determined by fluid-rock molecular interactions, alters interfacial properties like adsorbed film thickness and effective capillary pressure⁵². In our current model, wettability effects are implicitly captured within the generalized parameter and its associated correlations. For instance, the pronounced MMP reduction predicted for low systems likely encompass the combined impact of strong capillary confinement and the typically more significant fluid-rock interactions prevalent in oil-wet or mixed-wet tight rocks. Explicitly integrating wettability-dependent parameters, such as contact angle or adsorption energy, derived from molecular-scale insights could further refine the cell-to-cell simulation, particularly for predicting MMP in chemically heterogeneous reservoirs. This represents a valuable direction for future model development.

The second limitation can be attributed to the use of empirical relationships to calculate the relative permeability of oil and gas. The third limitation can be considered a result of the non-dynamic nature of the cell-by-cell model. In this model, after the fluids within a cell reach equilibrium, the excess volume is transferred to the next cell according to the relative permeability. The greater the number of cells, the closer the fluid motion modeling becomes to a dynamic state. Finally, an attempt is made to extrapolate the recovery value to an infinite number of cells. To investigate the porous media effect on MMP, it is necessary to perform cell-to-cell simulation for porous mediums with different characteristics and compare the results. For this purpose, in each permeability to porosity ratio, a five-point slim tube test simulation (calculation of ultimate ORF at five different pressures) has been carried out. It should be noted that the ultimate ORF should be calculated in an infinite number of cells. As a result, for different values of the permeability to porosity ratio, Fig. 11 is obtained. In this figure, at a constant temperature, with the permeability to porosity ratio reduction, the value of the MMP also decreases. The reason for this can be related to the reduction of the bubble point pressure (due to the reduction of the critical fluid properties) of the reservoir oil in the porous medium. In other words, with the bubble point pressure reduction, the miscibility of oil with the injected gas due to the reduction of the bubble point pressure of the final composition (the result of gas dissolution in the reservoir oil), the MMP will also decrease. On the other hand, with the permeability-to-porosity ratio reduction, the capillary pressure and the oil-gas IFT also increase. With the increase of capillary pressure and IFT between the two phases, it is expected that the MMP will increase. Therefore, the capillary pressure and interfacial tension increment, and the oil bubble point pressure reduction in the reservoir act in opposite directions. However, the results of the simulation show that the bubble point pressure reduction due to the critical properties reduction of the fluid overcomes the capillary pressure and the IFT increment, and the result of these two processes results in a decrease in the MMP with a reduction in the permeability to porosity ratio.

Fig. 11.

MMP versus the permeability-to-porosity ratio at different temperatures.

According to Fig. 11, the changes in the MMP are more significant for values ​​less than 10. This indicates that the lower the permeability to porosity ratio of a reservoir, the greater the influence of the reservoir rock on the MMP. This will be exacerbated in carbonated, tight, and shale reservoirs, where the use of traditional cell-to-cell simulations can have significant errors relative to the real situation. To investigate this issue, four models were compared with experimental slim tube data, which can be seen in Fig. 12. In this figure, the traditional cell to cell simulation (Bulk CCS), the modified model considering only the capillary pressure effect (Pc corrected CCS), the modified model considering only the critical properties shift CCS, and the modified CCS model considering the combined effect of capillary pressure and critical properties shift are compared with the slim tube test data. For better clarity, in another plot (Fig. 13), the MAPE of all models relative to the slim tube test data has been compared, according to which the developed model had the least error. The validity of the results of the modified PR-EoS based on permeability to porosity ratio has been indirectly examined, because no experimental method has been presented so far to validate the results of the modified equation of state, which is in terms of permeability and porosity. This indirect validation study was conducted by using the modified PR-EoS in a slim tube experiment simulation with a porous medium (that has a permeability to porosity ratio). Considering that the results obtained from using the modified PR-EoS are closer to the experimental data than the traditional PR-EoS, it can be concluded that the modified PR-EoS has appropriate computational validity (Figs. 12 and 13).

Fig. 12.

The comparison of all models in ORF prediction with slim tube test results.

Fig. 13.

The comparison of all model MPAE with slim tube test.

It is also important to examine how changes in the permeability-to-porosity ratio affect the ultimate oil recovery. For this purpose, a diagram of oil recovery according to the permeability to porosity ratio at different pressures is presented, as shown in Fig. 14. According to this figure, a reduction in the permeability-to-porosity ratio leads to an increase in ultimate oil recovery under constant pressures and immiscible conditions. Because as the pore size decreases, the critical properties decrease, which brings the conditions closer to miscibility conditions. The closer the gas injection conditions are to miscible condition, the higher the injection efficiency and, as a result, the higher the ultimate recovery. It is expected that the increase in the ultimate oil recovery due to the reduction in the permeability to porosity ratio at constant pressure will cause a decrease in the MMP, as shown in Fig. 11.

Fig. 14.

Ultimate oil recovery versus the permeability-to-porosity ratio for different pressures.

In this study, the effect of temperature on MMP was also investigated. For this purpose, several simulations were performed at different temperatures, and a diagram of the MMP in terms of temperature is shown in Fig. 15. According to the results obtained, it can be seen that with an increase in temperature, the MMP will also increase. This phenomenon can occur because of a decrease in the solubility of the two liquid and gas phases with increasing temperature. In other words, the solubility of gases in liquid has an inverse relationship with temperature, and with increasing temperature, the solubility of gas in the liquid decreases. This solubility decrement means that higher pressures are needed to dissolve a certain amount of gas in the liquid; therefore, the MMP will increase. Of course, it should be noted that this observation is true for hydrocarbon gases, but for carbon dioxide, this process can change at high temperatures. According to the studies of Sun and Li^47,53, in the low-temperature range, with an increase in temperature, the solubility of gas in the liquid is reduced, and as a result, the MMP will also increase. However, in the high-temperature range, with an increase in temperature, the extraction of the intermediate oil components by carbon dioxide increases because of the robust increase in the carbon dioxide density; therefore, the MMP decreases. Investigation of the changes in the MMP with temperature was performed for both free bulk fluid and confined fluid, and almost the same behavior was observed in both cases.

Fig. 15.

MMP variation with temperature for free bulk and confined fluid.

One of the investigations carried out in this modeling is the analysis of the composition of live oil produced from the slim tube. For this purpose, the intermediate oil components with the highest probability of exchange between the two phases in both the evaporation and condensation mechanisms were surveyed in the produced live oil. This investigation was performed under both immiscible (20 MPa) and miscible (31 MPa) conditions. The first is related to the immiscible condition, where the graph of the ratio of the mole fraction of each component in the produced oil to that of that component in the primary oil is drawn in terms of the pore volume injection (Fig. 16). Until the gas breakthrough, the mole fraction ratio of ethane to heptane remained constant. However, after the gas breakthrough, all components except ethane decreased slightly but remained unchanged even after the breakthrough. However, over time from the gas injection after the breakthrough (at 0.74 pore volume), the amount of fluid components in the output has deviated from the original oil conditions. Following this deviation, the concentrations of ethane and propane in the output oil increased. On the other hand, the amount of isobutane to heptane decreased from the initial amount in the reservoir oil. This means that the components of ethane and propane entered the oil from the injected gas, and this was the direction of the transfer of other components from the oil to the injected gas. Due to the nature of the transferred components, the produced oil is expected to be lighter than the initial oil in place. This lighter oil is one of the most favorable conditions resulting from gas injection into the reservoir. In this situation, although the gas injection was immiscible, the oil becoming lighter will enhance its movement toward the production point.

Fig. 16.

Ratio of the mole fraction of each component in the produced live oil to the mole fraction of that component in the primary oil in terms of pore volume injection under immiscible conditions, 20 MPa.

This investigation was carried out similarly under miscible conditions (31 MPa), and a graph of the ratio of the molar fraction of each component in the produced live oil to that of that component in the primary oil according to the pore volume injection is shown in Fig. 17. This figure shows the amount of each component at the beginning of the injection is equal to its amount in the primary oil. This is proof of the claim that at the beginning of the injection, the gas injection front has not reached the end of the slim tube, and the produced oil is the same as the original oil in place. However, before the gas breakthrough, it was observed that the number of all components in the produced oil had changed compared with the primary oil. In this way, both ethane and propane have started to increase, but other components have decreased compared with the original oil. The occurrence of this event before the gas breakthrough indicates that the gas miscibility process in the primary oil occurred earlier than the gas injection front and caused the composition of the primary oil to change before the gas breakthrough. This phenomenon justifies the importance of miscible conditions compared with immiscible conditions for gas injection.

Fig. 17.

Ratio of the mole fraction of each component in the produced live oil to the mole fraction of that component in the primary oil in terms of pore volume injection under miscible conditions, 31 MPa.

Changes in the components of the produced oil will continue after the gas breakthrough. The changes in more detail after the breakthrough are shown in Fig. 18. According to this figure, after the gas breakthrough, ethane and propane also showed an increasing trend similar to the immiscible state. In addition, n-pentane has a relatively decreasing trend under immiscible conditions. However, the behavior of the other oil components varies under immiscible conditions. According to Fig. 18, i-butane decreased after the gas breakthrough, but after some time, its trend changed from reduction to increment, and finally, it almost reached the initial amount of i-butane in the original oil in place. This behavior shows that during the process of gas injection using miscible conditions, i-butane entered the gas first from the oil, but after the injection continued, and because of the lighter oil in the slim tube, the i-butane transfer process changed. This process has also occurred for n-butane and i-pentane, but the final amount of these two components is lower than the initial amount in the original oil in place. Different conditions were established for the hexane and heptane components. After the gas breakthrough, the amount of these two components decreased; however, after some time, these two components entered the oil from the injected gas, and their final amount after injection of 1.2 pore volume was more than the amount in the original live oil.

Fig. 18.

Ratio of the mole fraction of each component in the produced live oil to the mole fraction of that component in the primary oil in terms of pore volume injection under miscible conditions, 31 MPa, after a breakthrough.

To compare immiscible and miscible conditions, it can be said that in immiscible conditions, light components such as ethane and propane are more abundant in the produced oil than in the primary live oil. On the other hand, the heavier components in the produced oil are reduced compared with the primary oil. Therefore, generally, the produced oil became lighter than the primary oil. However, under miscible conditions, light components such as ethane and propane have increased in the produced oil. In addition, intermediate components, such as n-butane, i-pentane, and n-pentane, finally decreased compared with primary oil. Components such as i-butane, hexane, and heptane have also remained constant or increased compared with the original oil. Therefore, it is necessary to use other properties of the produced oil, such as density and viscosity, to determine the interactions between the injected gas and the reservoir oil.

It should be noted that determination of the breakthrough time in the slim tube test must be related to the breakthrough time in the modeling. In the slim tube test, a sight glass is placed after the slim tube outlet, which can be used to determine the exact breakthrough time using image analysis. At pressures of 20 and 31 MPa, the breakthrough times were measured to be about 0.098 and 0.81 pore volume respectively. This is while this time is calculated to be equal to 0.11 and 0.74 pore volume in the developed cell-by-cell model. This indicates the closeness of the experimental and modeling values for the breakthrough time.

A graph of the density and viscosity ratios relative to their initial values, according to the pore volume injection, is shown in Fig. 19. According to this figure, before the injection gas breakthrough, the density and viscosity of the oil produced under immiscible conditions were equal to the density and viscosity of the primary live oil. This means that the gas injection front has surpassed the gas miscibility front, and the gas injection has not changed the primary oil. However, under miscible conditions, this behavior changes with the oil’s density and viscosity beginning to decrease even before the gas reaches the outlet of the slim tube. This means that under miscible conditions, the miscibility front has overtaken the gas injection front, and before the gas breakthrough, the interactions between oil and gas have caused a change in the composition of oil and gas. After gas breakthrough under both immiscible and miscible conditions, the density and viscosity of the produced oil decreased compared with the original oil. However, under immiscible conditions, the density and viscosity of the produced oil first decrease, then increase and finally decrease. This event indicates that under immiscible conditions, the exchange process of different components between oil and injected gas changes in several stages. By comparing these results with Fig. 16, it can be concluded that under immiscible conditions, ethane is not exchanged between oil, and injected gas and the changes in propane are very small, but other intermediate components will decrease first and then increase. This conclusion is fully consistent with the density and viscosity diagram shown in Fig. 19.

Fig. 19.

Graph of density and viscosity relative to the initial value of the ratio of produced live oil according to the pore volume injection for two immiscible (20 MPa) and miscible (31 MPa) conditions.

The mentioned changes are completely different for miscible conditions. Under miscible conditions after gas breakthrough, the density and viscosity of the produced oil are still decreasing compared with the original oil. Therefore, in the competition between increasing the amount of ethane and propane and decreasing i-butane, hexane, and heptane, the effect of increasing ethane and propane is much greater than that of decreasing isobutane, hexane, and heptane. As a result, the density and viscosity of the produced oil always decrease compared to the original oil. Of course, this issue is also quite clear in Fig. 18. In this graph, the changes in ethane and propane are much higher than those in i-butane, hexane, and heptane. Therefore, the produced oil is expected to be lighter than the original in situ oil, as shown in Fig. 18.

Conclusions

This study estimates the minimum miscibility pressure (MMP) by integrating laboratory slim-tube test with a modified cell-to-cell simulation (MCCS). Its primary contribution is a reliable computational tool adapted for porous media, achieved by modifying the Peng-Robinson equation of state to incorporate permeability and porosity, refining the fluid transfer criterion to account for gas channeling, and including capillary pressure in phase behavior calculations. These key corrections are integrated using the permeability-to-porosity ratio as a scaling parameter, moving beyond standard CCS models. Rigorously validated against experimental data, this methodology leads to a crucial engineering insight: it demonstrates that lower permeability-to-porosity ratios, characteristic of tight formations, substantially reduce the MMP. This finding indicates that miscibility can be achieved more readily in tight reservoirs than previously assumed, directly influencing gas injection project design. Finally, the main conclusions of this research can be expressed as follows:

• By performing a slim tube test at five pressure points of 20.00, 22.12, 24.13, 28.96, and 31.0 MPa, the results show that the MMP of the injected gas and reservoir is approximately 25.28 MPa.

• In this study, we attempted to modify the cell-to-cell simulation to apply the porous media effect in the slim tube test. The applied modifications include Peng-Robinson EOS modification for a porous medium by considering its porosity and permeability, modification of the equations of critical pressure and temperature shifts by the porosity and permeability of the porous media, application of the capillary pressure effect in VLE calculations, and modification of fluid transfer criteria within the cell-to-cell model after the occurrence of gas breakthrough.

• Simulations of the slim tube using modified cell-by-cell simulation (MCCS), revealed that the ultimate oil recovery increases with the number of modeling cells. Therefore, it is necessary to calculate the ultimate oil recovery for different numbers of cells and constant pressure and obtain the ultimate oil recovery for an infinite number of cells through extrapolation. In this case, the dispersion effect is zero.

• Validated against experimental data, the modified CCS (MCCS) model demonstrates good predictive accuracy. The results show that it predicts the ultimate oil recovery at each pressure point very close to the value obtained from the slim tube test, with a Mean Absolute Percentage Error (MAPE) of 5.52%. The calculated mean error of + 5.52% suggests a slight, consistent overestimation, as evidenced by the MMP obtained from the slim tube test (25.28 MPa) and the MCCS estimate (26.11 MPa), which is approximately 0.83 MPa higher. The 95% confidence interval for this mean error (−0.73% to + 11.77%) indicates this overestimation provides a practical safety margin on average. Consequently, the MCCS serves as a reliable and efficient tool for screening and preliminary design in gas injection projects, though the small difference from laboratory values highlights the value of continuing its refinement in future studies.

• The investigation of temperature’s effect on MMP shows that MMP increases with temperature. This is because, as the temperature rises, the solubility of gasses in oil decreases, requiring higher pressure to achieve complete mixing of the two phases.

• To investigate the effect of the porous medium on the MMP, this parameter was calculated for different permeability-to-porosity ratios. The results show that the MMP decreases with permeability to porosity reduction. The trend is gentle for ratios around 10, but for ratios below 10, the reduction trend is robust. Therefore, in tight oil reservoirs, achieving miscibility is easier. Although due to the permeability to porosity ratio in the slim tube (around 10), the error of the MMP has an acceptable value, for the simulation of this test in tight reservoirs and shale reservoirs, the proposed algorithm is suggested.

• One of the most important issues raised in gas injection processes is the existence of evaporation and condensation mechanisms. In this study, to investigate this issue, the composition of the live oil output from the slim tube was analyzed using a cell-to-cell simulation. In immiscible conditions, light components such as ethane and propane are more abundant in the produced oil than in the primary oil, and on the other hand, the heavier components in the produced oil are reduced compared to the initial. Therefore, in general, the produced oil is lighter than the primary oil. However, under miscible conditions, the concentration of light components such as ethane and propane increases in the produced oil. In addition, intermediate components, such as n-butane, i-pentane, and n-pentane, have finally decreased compared with primary oil. Components such as i-butane, hexane, and heptane have also remained constant or increased compared with the initial.

• According to the results of the produced oil properties, such as composition, density, and viscosity, the cell-to-cell simulation can also be used for monitoring the component exchanges and mechanism (vaporization and condensing) determination of gas injection in immiscible and miscible conditions.

Based on the results presented in this article, despite various modifications, the predicted value of the MMP with the laboratory value is still in error. Therefore, it is suggested that future work consider objectives, such as modifying the bundle of tube model for porous media, addressing the phenomenon of oil trapped in isolated pores, exploring asphaltene deposition in heavy oil reservoirs, and investigating the Marangoni effect. Additionally, while this study used a hydrocarbon gas for miscible gas injection, it is recommended to repeat this study with other injectable gasses, such as carbon dioxide, to evaluate their properties in flooding⁵⁴.

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

Ali Safaei: Performing the experiments and developing the model and writing the manuscript.Masoud Riazi: developing the model and writing the manuscript and superviser.Jafarzadegan: Supervising the modeling.Naserpour: Supervising the experiments.

Funding

The author received No Funding for this work.

Data availability

The datasets generated and/or analyzed during the current study are not publicly available but are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.