A Systematic Computational Study of Oil Displacement Processes in Terrigenous and Cavernous-Fractured Porous Media Using Surfactant Solutions

Abstract

This paper presents the results of a numerical simulation of oil displacement from models of terrigenous and cavernous-fractured media using solutions of the anionic surfactant (sodium laureth sulfate). The surfactant concentration was varied from 0 to 0.1 wt.%. The simulations employed a mathematical model for the flow of immiscible liquids based on the VOF method. The model incorporated experimentally measured interfacial tension coefficients and wettability parameters for the surfactant solutions. The results demonstrate that increasing the surfactant concentration enhances the oil displacement coefficient: by 15% for the terrigenous model and by 19% for the cavernous-fractured model compared to water flooding (at 0 wt.% surfactant), achieving a maximum at a concentration of 0.1 wt.%. The influence of potential mechanisms leading to the improved oil displacement coefficient during surfactant solution injection was investigated. It was established that at a fixed displacement rate, the addition of the surfactant causes a local increase in the generalized capillary number by a factor of approximately 3.7. This is identified as the primary mechanism for the observed enhancement of the oil displacement coefficient in this case. The data obtained in this study can be used for further improvement of surfactant flooding technologies for enhanced oil recovery.

1. Introduction

The share of hard-to-recover hydrocarbon reserves is increasing relative to primary and secondary oil recovery, necessitating the development of new enhanced oil recovery (EOR) methods. During oil production, the use of primary and secondary recovery methods allows for the extraction of approximately 20–45% of the original oil in place (OOIP), depending on the properties of the oil and the reservoir rock [1,2]. The remaining oil can be targeted through the application of Enhanced Oil Recovery (EOR) techniques [2].

Chemical EOR methods include surfactant flooding. Currently, EOR techniques based on the use of surfactant solutions and polymer solutions are employed. Surfactant flooding is a complex, multi-stage process. Surfactants are added to the water used for reservoir flooding [3]. The use of these chemicals reduces the capillary forces that trap oil within the rock pores. Furthermore, surfactants can adsorb onto the rock surface, altering its wettability towards a more hydrophobic state [4,5,6], which enables water to displace oil more effectively. Oil reservoirs possess unique fluid and rock properties; therefore, tailored chemical systems must be developed for each application, considering salinity, temperature, and rock type. The specific chemical agents used, their concentrations in the injected slugs, and the slug sizes depend on the particular fluid and rock properties, as well as economic considerations.

The reservoir type is critically important not only for selecting enhanced oil recovery methods but also for technologies related to carbon dioxide sequestration and hydrogen storage. The sequestration of carbon dioxide in depleted reservoirs, on the one hand, contributes to the removal of greenhouse gases from the atmosphere and, on the other hand, can be an effective method for enhancing oil recovery. For the effective utilization of oil and gas reservoirs, they must meet specific requirements [7,8]. They should possess high porosity and permeability; such reservoirs are typically sandstones, which must be located at sufficient depth to maintain carbon dioxide in a liquefied or supercritical state. Furthermore, it is essential that highly permeable reservoir layers are capped by impermeable layers to prevent gas leakage. Even more stringent requirements are imposed on underground reservoirs for hydrogen storage [9,10]. Underground Hydrogen Storage (UHS) in depleted oil and gas formations is considered a promising technology for long-term energy storage within the hydrogen economy. Since hydrogen is highly mobile, the reservoir matrix must be virtually impermeable.

Surfactant flooding is a complex process that requires detailed laboratory testing to support the design of a field project. As evidenced by field applications, it shows excellent potential for improving the recovery of low- and medium-viscosity oils [11].

The simulation of surfactant flooding at the reservoir scale is performed using various simulators. UTCHEM (University of Texas Chemical Flooding Simulator) is a research simulator that uses the finite difference method. Reference [12] proposed a Black oil model incorporating microemulsion behavior to simulate the surfactant-gas flooding process. STARS (CMG) is a commercial compositional simulator capable of modeling surfactant flooding via its “miscible flood” option and chemical reaction module, utilizing a volume balance approach. A study [13] using this simulator investigated continuous flooding with a proprietary surfactant in the Bentiu oil reservoir (Sudan). The results demonstrated that surfactant flooding increased the oil recovery factor to 70%, compared to only 33% for waterflooding. GEM (CMG) is a more advanced compositional simulator that can also account for complex phase behavior. Reference [14] conducted a numerical study based on experimental data for applying chemical EOR methods in carbonate reservoirs. ECLIPSE (Schlumberger) addresses a full spectrum of reservoir simulation challenges, including finite-difference models for black oil, dry gas, compositional gas-condensate, thermal heavy oil, and streamline models. Its advanced functionalities can be adapted for surfactant flooding simulations, although it is less specialized than STARS or UTCHEM. A study [15] investigated numerical errors during surfactant flooding modeling in the commercial ECLIPSE simulator. It was shown that non-physical numerical artifacts may arise, particularly on coarse grids, with special attention required to the properties of the miscible flow. Therefore, thorough studies of both numerical and physical models may be necessary to mitigate instabilities and accurately predict the impact of surfactant injection on oil production.

Furthermore, molecular dynamics simulation [16] is employed to describe the microscopic interactions of surfactant molecules (AES/CAB) at the crude oil/formation water interface in the presence of salt ions.

A novel modeling approach involves neural network methods. The study [17] investigated the impact of various injection schemes on flooding efficiency using the example of oil and gas reservoirs in the Ordos Basin. This modeling technique has the potential to become an effective tool for optimizing enhanced oil recovery (EOR) strategies.

Given that conducting filtration experiments to select the appropriate surfactant type and concentration is a time-consuming and costly process, this paper explores the feasibility of using numerical modeling based on the VOF (Volume of Fluid) method to screen different surfactant types. The model utilizes experimentally determined values of interfacial tension coefficient and contact angle. Unlike the simulation methods reviewed earlier, the VOF approach offers the additional capability to analyze velocity fields and the displacement front propagation within porous media. A particularly significant aspect of this work is that, for the first time, numerical simulation has been used to investigate the specifics of oil displacement by surfactant solutions from two distinct types of hydrocarbon reservoirs: terrigenous and cavernous-fractured.

2. Numerical Methodology

This work presents, for the first time, a systematic numerical investigation of oil displacement processes from models of terrigenous and cavernous-fractured media using AES surfactant solutions. The simulation of oil displacement from porous media by surfactant solutions was performed using the VOF (Volume of Fluid) method [18] without affect adsorption surfactant on the rock. This method is designed for modeling immiscible, multi-component liquid–liquid flow by solving a set of momentum equations and utilizing the volume fraction of each phase. The volume fraction of the displacing fluid, α, and the volume fraction of oil, β, are introduced. Given that α + β = 1, it is sufficient to use only the volume fraction of the displacing fluid, α. In this case, the density and viscosity of the mixture, which are used in the Navier–Stokes equations, are defined as follows:

$$\mathsf{\rho }={\mathsf{\rho }}_1\mathsf{\alpha }+(1-\alpha ){\mathsf{\rho }}_2,$$

(1)

$$\mathsf{\mu }={\mathsf{\mu }}_1\alpha +(1-\alpha ){\mathsf{\mu }}_2,$$

(2)

where $\mathsf{\rho }$₂, μ₂—density and viscosity oil, a $\mathsf{\rho }$₁, μ₁—density and viscosity of the displacing fluid.

$$\partial \mathsf{\rho }/\partial \mathrm{t}+\nabla (\mathsf{\rho }·\overrightarrow{\mathrm{V}})=0,$$

(3)

where $\overrightarrow{\mathrm{V}}$ is the velocity vector of the mixture, determined by solving the momentum equation:

$${\displaystyle \frac{\partial }{\partial t}}(\mathsf{\rho }\overrightarrow{\mathrm{V}}) + \nabla ·(\mathsf{\rho }\overrightarrow{\mathrm{V}}\overrightarrow{\mathrm{V}}) = -\nabla \mathrm{p} + \nabla ·[\mathsf{\mu }(\nabla \overrightarrow{\mathrm{V}}+ \nabla {\overrightarrow{\mathrm{V}}}^T)] + \overrightarrow{F_s},$$

(4)

where p is the static pressure of the mixture, $\overrightarrow{F_s}$ is the volumetric force vector arising from capillary forces.

The motion of the interface during oil displacement is modeled by solving the following advection equation:

$${\displaystyle \frac{\partial \alpha }{\partial t}}+ \nabla ·(\alpha \overrightarrow{\mathrm{V}})=0,$$

(5)

Capillary forces are crucial when oil is displaced from a porous medium. To model surface tension, the Continuum Surface Force (CSF) algorithm proposed by Brackbill et al. [19] was used. In this method, surface tension effects are modeled by adding a source term to the momentum equation:

$$\overrightarrow{F_s}=\sigma k\nabla \alpha$$

(6)

where σ is the interfacial tension coefficient, k is the interface curvature, defined as:

$$k= \nabla \left({\displaystyle \frac{\overrightarrow{n}}{\left|\overrightarrow{n}\right|}}\right),$$

(7)

where $\overrightarrow{n}$ is the interface normal vector, which is computed within the entire computational domain as:

$$\overrightarrow{n}=\nabla \mathsf{\alpha }$$

(8)

At the walls of the computational domain, the normal vector is defined by the following expression:

$$\overrightarrow{n}= {\overrightarrow{n}}_{\mathrm{w}} \cos (\mathsf{\theta }) + {\overrightarrow{\tau }}_w\sin (\mathsf{\theta }),$$

(9)

where θ the contact angle at the wall, ${\overrightarrow{n}}_w$ is normal to the wall vector, ${\overrightarrow{\tau }}_w$ is tangent to the wall vector.

The system of nonlinear differential Equations (3)–(5) was solved using the Finite-Volume Method (FVM) [20]. The coupling between the velocity and pressure fields was implemented using the SIMPLEC (Semi-Implicit Method for Pressure-Linked Equations Consistent) algorithm. The PRESTO (PREssure Staggering Option) scheme was employed for discretizing the pressure term to ensure continuity balance. A second-order implicit scheme was used for the discretization of the transient terms in the transport equations. Convective terms in the Navier–Stokes equations were approximated using a second-order central differencing scheme. The advection equation was solved using a TVD (Total Variation Diminishing) scheme with the HRIC (High-Resolution Interface Capturing) limiter. This numerical approach has been previously validated by the authors in several studies [21,22,23].

3. Problem Statement and Boundary Conditions

A computational study of oil displacement by surfactant solutions was conducted in a two-dimensional isothermal formulation. The study considers two models of porous media: a terrigenous model and a cavernous-fractured model.

Figure 1a shows the 2D computational geometry of the terrigenous porous medium used in the simulations. The computational domain is a rectangular parallelogram section containing a porous medium modeled by random packing of spheres of various sizes. Initially, the void space is saturated with oil. The dimensions of the domain are 5.4 mm in length and 1.87 mm in width. This geometry is formed by the random packing of circles with three distinct diameters: a minimum of approximately 14 µm, a mean of 87 µm, and a maximum of approximately 120 µm. The minimum channel size formed by the packing is 7.5 µm. The porosity of the domain is 30%, and the permeability is 52 mD. The computations utilized a mesh consisting of 400,000 cells. This mesh resolution was selected based on a grid sensitivity analysis. Fragments of meshes with different resolutions are presented in Figure 2a. The influence of mesh resolution was assessed by the pressure drop during single-phase water flow through the domain at a fixed inlet velocity of 1.25 × 10⁻³ m/s. The results are presented in

Table 1. Studying of the detailing of computational grid the terrigenous model.

Number of Grid Cells	Kdisp, %	Deviation, %	ΔP, Pa	Deviation, %
400,000	56.9	4.9	570.98	5.9
750,000	52.9	0.9	581.35	4.0
2,800,000	52.0	0	605.10	0

. As can be seen, the pressure drop obtained on the 400,000-cell mesh differs by 6% from the value for the finest mesh, which is considered sufficient accuracy for the simulations and allows for a reduction in computational time.

Figure 1b shows the 2D computational geometry of the cavernous-fractured medium used in the calculations. The geometry of the computational domain is a rectangular parallelogram section with a fractured medium. The dimensions of the domain are 7.45 mm in length and 6.8 mm in width. The minimum channel width was 50 µm, the average width was 110 µm, and the maximum cavity cross-section was 500 µm. A detailed description of the geometry and its generation process is provided in reference [21]. For this cavernous-fractured geometry, a computational mesh with a total of 415,000 cells was generated. Similar to the porous medium model, a grid sensitivity study was performed for the fractured medium model, leading to the selection of the mesh with 415,000 cells. Fragments of meshes with different resolutions are presented in Figure 2b. A comparison of the pressure drop for meshes of varying resolution was conducted analogously to the terrigenous rock model (see

Table 2. Study of the detailing of computational grid the cavernous-fractured model.

Number of Grid Cells	Kdisp, %	Deviation, %	ΔP, Pa	Deviation, %
415,000	47.1	4.1	38.89	3.3
1,400,000	45.0	2.0	39.92	0.72
5,100,000	43.0	0	40.21	0

). For this model, the difference between the selected mesh and the finest mesh is 3.3%.

At the initial moment, the pore space is saturated with oil. The velocity of the displacing fluid was specified at the inlet of the computational domain. A no-slip boundary condition was set at the walls of the domain. The value of the contact angle, determined experimentally, was specified at the solid walls. The methodology for measuring the contact angle is described in detail in Section 3. A Neumann boundary condition was set at the outlet of the computational domain.

The simulations were performed on a 32-core processor with a clock speed of 4 GHz. The typical time step, determined from the Courant–Friedrichs–Lewy (CFL) condition, was approximately 10⁻⁶ s. The displacement process was simulated for a total duration of 10 s of physical time. Over this period, approximately 3 pore volumes were injected. The typical computational time for a single simulation run under these conditions was 10 days.

During the simulation, the values of the oil displacement coefficient and the pressure drop during the injection of the displacing fluid were determined. The oil displacement coefficient is defined as the ratio of the volume of displaced oil to the initial volume of oil in the pore/fractured space.

4. Property Displacing Fluids

Preliminary laboratory investigations of the properties were conducted for the surfactant solutions under consideration. The study utilized solutions of commercial-grade surfactants. Solutions of sodium laureth sulfate (AES) at various concentrations were examined. Sodium Laureth Sulfate 70% (Tainolin, AES-70-2NC) is an anionic surfactant with the chemical formula: C₁₂H₂₅(OCH₂CH₂)₂OSO₃Na. It is widely known as a primary ingredient in the production of cosmetics and household chemicals, exhibits low skin irritation, and forms stable foam. It is a paste-like product, ranging from colorless to light yellow, representing an aqueous solution of sodium salts of C₁₂–C₁₄ aliphatic alcohol dioxyethylated sulfates of natural or synthetic origin. The active matter content is 70.0 ± 2.0 wt.%.

Systematic measurements of physical properties were performed for all surfactant solutions used in the study. The values of the dynamic viscosity coefficient, interfacial tension coefficient, and contact angle in the three-phase system displacing agent/oil/rock were measured. Berea sandstone (Berea Clear Birmingham Buff, Cleveland, USA) was used as the model rock sample, with an average porosity of 21% and permeability of 50 mD.

The viscosity of the water, oil, and aqueous AES solutions was measured using a Brookfield DV2T rotary viscometer (ULA spindle). The reproducibility limit of the viscosity measurements does not exceed 0.5%, and the limit of allowable reduced error for the ULA spindle is ±1%. The measurement technique is described in detail in [24]. The measurements were carried out at 25 °C. The results of the dynamic viscosity coefficient measurements showed that for the considered surfactant concentrations, the viscosity values change only slightly compared to water. For instance, at a concentration of 0.1%, the viscosity is 2% higher than that of water, which is comparable to the measurement error. The influence of the oil-to-displacing fluid viscosity ratio on the oil displacement efficiency was investigated in our previous study [25]. That work examined viscosity ratios ranging from 1 to 400, while the viscosity of the displacing fluid was held constant. A significant change in the viscosity ratio—specifically, from 10 to 40—was found to alter the oil displacement coefficient by approximately 10%. In the present study, the oil-to-displacing agent viscosity ratio is 47. Consequently, a 2% variation in the viscosity of the displacing agent is expected to have a negligible effect on the oil displacement coefficient. For this reason, a constant viscosity value is adopted for the displacing agents considered herein.

The interfacial tension σ of the boundary of all displacing agents oil and the wetting angle θ were investigated using the technique, which was employed earlier [26]. A CoreLab TEMCO IFT-820-P hanging drop surface-tension measurement system was used to measure interfacial tension. The contact angle and interfacial tension were both measured using the DropImage Advanced software (version 2.5) (Ramé-Hart Instrument Co., Succasunna, NJ, USA). All surface tension and contact-angle measurements were performed at atmospheric pressure and 25 °C. Under controlled laboratory conditions, a measurement accuracy of ±0.5 was achieved for the contact angle and ±0.2 mN/m for the interfacial tension.

An investigation was conducted on the influence of surfactant concentration on the contact angle in the “AES/oil/rock” system. For these studies, crude oil with a density of 874 kg/m³ and a viscosity of 47 mPa·s was used. Photographs of oil droplets in multicomponent solutions with different AES concentrations on the rock are shown in Figure 3.

The concentration dependence of the contact angle of oil on the rock was plotted (see Figure 4). It was found that an increase in the AES concentration in water leads to a decrease in the contact angle; however, the relationship is non-monotonic. At low surfactant concentrations, a hydrophilizing effect is generally observed, while a hydrophobizing effect occurs at high concentrations. This non-monotonic behavior of the contact angle is attributed to the formation of stable surfactant micelles in the solution at high surfactant concentrations (0.1 wt.%) that exceed the critical micelle concentration (CMC). Consequently, the concentration of surfactant molecules at the rock/oil interface effectively decreases, leading in turn to a slight reduction in the contact angle.

For the surfactant solutions used, the interfacial tension (IFT) in the “displacing fluid-oil” system was investigated. The interfacial tension was determined from the geometric dimensions of a pendant oil droplet in the displacing fluid, based on the density gradient. Figure 5 presents photographs of pendant oil droplets in solutions with different AES concentrations. The dependence of the “surfactant solution-oil” interfacial tension on the AES concentration was obtained. Figure 6 shows that the influence of AES on the IFT begins at low concentrations: the addition of 0.01% reduces the IFT by a factor of two. The effect of the AES additive on the IFT is non-linear, and a further increase in concentration has a less pronounced effect. For instance, increasing the concentration from 0.01% to 0.05% reduces the interfacial tension coefficient by a factor of 1.5, while an increase from 0.05% to 0.1% leads to a further 13% reduction.

The data on interfacial tension and wettability of the AES solutions obtained from the laboratory experiments were used as input parameters for the numerical simulations, the results of which are presented below.

5. Results and Discussion

5.1. Investigation of the Effect of Surfactant Concentration on the Oil Displacement Process in Porous Media

A numerical study was conducted on the process of oil displacement from models of terrigenous and cavernous-fractured media using a 70% solution of Sodium Laureth Sulfate (AES) at various mass concentrations. In the simulations, the surfactant mass concentration was varied within the range of 0.01% to 0.1%. In real reservoir conditions, the displacement velocity varies depending on reservoir permeability and injection pressure. However, at the micro-level, it is difficult to operate with injection pressure. Therefore, we employed a setup in which the injection rate was specified. This study utilized a fixed inlet rate to specifically evaluate the impact of changes in the interfacial tension coefficient and contact angle when using surfactant solutions. This approach allows the results to be generalized using the capillary number. A fixed velocity of the displacing agent, 1.25 × 10⁻³ m/s. was specified at the inlet of the computational domain. This velocity value corresponds to a capillary number of 4.17 × 10⁻⁵, as determined for water. The capillary number in the calculations was determined by the formula: Ca = Vμ/σ, where μ and σ are the viscosity and the interfacial tension coefficient of the displacing liquid, respectively.

The simulations utilized experimentally determined values of the contact angle at the AES surfactant/oil/solid rock interface, the interfacial tension coefficient, and the viscosity of the AES surfactant solutions as functions of mass concentration. The description of the experimental measurement techniques and the results are presented in Section 4.

During the numerical investigation of the oil displacement process from the porous medium, distribution patterns of the displacing agents (water and AES surfactant solutions) at different time instances were obtained (see Figure 7). As can be seen from Figure 7, after water flooding, significant regions of residual oil remain along the lower and upper boundaries of the model. In contrast, when AES surfactant solutions are used as the displacing agent, the areas saturated with oil are reduced compared to water, and this effect intensifies with increasing surfactant mass concentration. An increase in the surfactant concentration in the solution reduces the interfacial tension at the oil/displacing agent boundary, which in turn leads to a decrease in the capillary forces trapping the oil. This results in the displacement of additional, previously unswept areas within the model rock.

Furthermore, the velocity field distributions within the porous medium were obtained from the calculations. As an example, Figure 8 presents the velocity field distributions for different surfactant concentrations at the end of the displacement process (15 s). Analysis of the velocity fields for water and various surfactant concentrations shows that the use of surfactants alters the velocity field. Unlike the case with water, where a single large flow channel forms, the surfactant solutions exhibit additional flow channels distributed across the width of the computational domain.

The calculations yielded the dependencies of the oil displacement coefficient and the pressure drop on the displacement time for water (the base displacing agent) and for AES surfactant solutions with various mass concentrations. Furthermore, based on this data, the dependencies of the oil displacement coefficient ($K_{disp}$) and the pressure drop on the surfactant concentration were plotted. The oil displacement coefficient is defined as the ratio of the cumulative oil volume produced from the model at a given time i ($V_i$) to the initial oil volume in the model ${(V}_0)$: $K_{disp}=V_i/V_0$. These dependencies are presented in Figure 9 and Figure 10, respectively. The results demonstrate that using AES surfactant solutions as the displacing agent leads to an increase in the oil displacement coefficient compared to water. The comparison of the oil displacement coefficients was performed under steady-state flow conditions, when the coefficient values stabilized and no longer changed with time, specifically within the range of 9 s to 10 s (see Figure 9a). The oil displacement coefficient increases with the mass concentration of AES surfactant. For instance, at a concentration of 0.01%, the oil displacement coefficient increases by 5.6% relative to water (see Figure 9b). An increase in the surfactant concentration from 0.01% to 0.05% results in a 7% rise in the displacement coefficient. The oil displacement coefficient for a concentration of 0.1% is 15% higher than that for water. This effect is attributed to the significant influence of the AES surfactant on the interfacial tension coefficient at the displacing agent/oil boundary, the value of which changes with increasing surfactant concentration. As shown in Section 3, the interfacial tension value decreases by a factor of 3.6 compared to water for an AES surfactant mass concentration of 0.1%.

Figure 10a shows the dependence of the pressure drop on the displacement time in the terrigenous rock model, while Figure 10b presents the dependence of the steady-state pressure drop value in the computational domain under stabilized flow conditions. For the surfactant solution with a mass concentration of 0.01%, the pressure drop value is 20% higher than that for water. As the AES surfactant concentration increases, the pressure drop decreases compared to water. Specifically, for an AES concentration of 0.1 wt.%, the steady-state pressure drop is 13% lower than for water. This can be explained by an increase in the number of flow channels available to the displacing agent, as the viscosity of the AES surfactant solution is lower than that of the oil. The pressure fluctuations observed in the graphs can be attributed to the fact that the total pressure, shown in the graphs, is the sum of the capillary and hydrodynamic pressure components. Hydrodynamic pressure is generally dependent on fluid viscosity and the average flow velocity. Capillary pressure is a function of interfacial tension and wettability. Consequently, the total pressure exhibits a complex dependence on surfactant concentration. At low surfactant concentrations, the displacement efficiency increased slightly, leading to a minor expansion of the flow region; however, the capillary pressure had not yet decreased sufficiently. As a result, the total pressure increased at these low concentrations. At higher surfactant concentrations, the reduction in capillary pressure becomes more pronounced, leading to an overall decrease in total pressure.

5.2. Investigation of the Effect of Surfactant Concentration on the Oil Displacement Process in a Cavernous-Fractured Medium

A numerical study of oil displacement processes from a cavernous-fractured medium by AES surfactant solutions at various mass concentrations was conducted. A fixed velocity of the displacing agent, 1.25 × 10⁻³ m/s. was specified at the inlet of the computational domain. This velocity value corresponds to a capillary number of 4.17 × 10⁻⁵, as determined for water. The results of the simulations yielded the distribution patterns of oil and the displacing agents (water and AES surfactant solutions at different concentrations) at various time instances during the displacement process (see Figure 11). Due to the reduction in the interfacial tension coefficient at the surfactant solution/oil boundary, the displacement front expands and the oil displacement coefficient from the cavernous-fractured medium increases compared to waterflooding. The figure shows that during displacement by surfactant solutions, additional channels and cavities within the computational domain are involved, unlike waterflooding, where only the central part of the domain is effectively swept. As the surfactant concentration increases, the swept zone within the computational domain expands, and the breakthrough time of the displacing fluid from the fractured model also increases. Changes in the velocity field distribution within the domain are also observed, as can be seen in Figure 12, which presents the velocity field distributions for water and different surfactant concentrations at a time of 4 s.

The calculations yielded the dependencies of the oil displacement coefficient and the pressure drop on the displacement time for water and for AES surfactant solutions with various mass concentrations. These dependencies are presented in Figure 13a and Figure 14a, respectively. Based on the obtained data, the dependencies of the oil displacement coefficient and the pressure drop under steady-state flow conditions were plotted, as shown in Figure 13b and Figure 14b, respectively. Analysis of the quantitative results showed that for a mass concentration of 0.01%, the increase in the oil displacement coefficient relative to water was less than 0.5%. As the concentration increased to 0.05%, the oil displacement coefficient increased by 15% compared to water. A further increase in concentration to 0.1% led to an additional 4% increase in the oil displacement coefficient relative to the value obtained for the 0.05% concentration.

The study also investigated the influence of surfactant concentration on the pressure drop in the cavernous-fractured medium during oil displacement. The dependence of the steady-state pressure drop at the end of the displacement process on the surfactant concentration is non-monotonic. For a concentration of 0.01%, the pressure drop is 8% higher than for water. As the surfactant concentration increases to 0.05%, the steady-state pressure drop decreases and becomes 5% lower than the value obtained for water. With a further concentration increase to 0.1%, the pressure drop is 7% lower than the value for water. Pressure fluctuations during the displacement process are associated with the detachment and expulsion of small oil droplets from pore cavities and the computational domain. This process leads to changes in capillary pressure, which, in turn, cause fluctuations in the total pressure.

In conclusion, let us discuss the influence of potential mechanisms that could lead to the increase in the displacement coefficient (K_disp). During surfactant injection, the reduction of Interfacial Tension (IFT) plays a key role in enhancing oil recovery. The interfacial tension data presented in Figure 6 show that the IFT coefficient decreases monotonically with increasing AES concentration. This unambiguously contributes to the increase in K_disp. As can be seen in Figure 15a, K_disp depends monotonically on the IFT coefficient (σ). Another factor influencing the displacement coefficient is the change in wettability and the contact angle ($\theta$). Improved wettability of surfactant solutions promotes better detachment of oil from the rock walls and, consequently, leads to an increase in K_disp. Figure 15b shows the dependence of K_disp on $cos\left(\theta \right)$ for the studied AES solutions. In this case, the relationship is non-monotonic. This is primarily explained by the non-monotonic dependence of $\theta$ on the surfactant concentration (see Figure 4). At low surfactant concentrations, wettability improves, while at very high concentrations, it slightly decreases, which should contribute to a reduction in K_disp. However, at high surfactant concentrations, the IFT is significantly reduced, which promotes increased oil recovery. Thus, the reduction in IFT compensates for the decrease in the contact angle $\theta$, and K_disp increases. Besides these two factors, oil displacement is influenced by the viscosity of the displacing fluid. The higher the viscosity of the displacing agent, the higher the displacement coefficient. However, as mentioned above, the effect of the AES surfactant additive on the solution viscosity was so insignificant that this factor was not considered in the calculations. The most comprehensive criterion determining the increase of oil displacement coefficient from porous media and accounting for the influence of interfacial tension, wettability, and viscosity is the generalized capillary number: ${Ca}^{\ast }=V\mu /\left(\sigma cos\left(\theta \right)\right)$, where $\mu$ is the viscosity of the displacing liquid, $V$ is the flow velocity, $\sigma$ is the interfacial tension coefficient, and $\theta$ is the contact angle. Using the available experimental and calculated data, the dependence of K_disp on ${Ca}^{\ast }$ was plotted. This dependence is shown in Figure 15c. It is evident that for both reservoir models, K_disp increases monotonically with increasing ${Ca}^{\ast }$. Therefore, at a fixed velocity of the displacing medium, the action of the surfactant additive leads to a local increase in the capillary number by almost a factor of four. This is the primary mechanism for the increase in K_disp in this case. It should also be noted that at a fixed ${Ca}^{\ast }$ value, K_disp for the cavernous-fractured rock model is lower than for the terrigenous model. This indicates that, all other factors being equal, the effect undoubtedly depends on the geometry of the pore space.

The key factors governing enhanced oil recovery during surfactant flooding are improved wettability and reduced interfacial tension. Undoubtedly, both factors positively impact displacement efficiency in both terrigenous and cavernous-fractured reservoir models. However, the degree of their influence differs. In the terrigenous reservoir model, the most significant factor is the surfactant-induced wettability improvement with increasing concentration. This promotes more effective stripping of oil films from rock grain surfaces. For the cavernous-fractured reservoir model, the greatest effect is achieved through the reduction in interfacial tension observed at higher surfactant concentrations. This leads to a decrease in capillary pressure, thereby mobilizing capillary-trapped oil from the cavities.

As highlighted in the literature review [4,5,6], surfactant adsorption on the rock surface is a significant factor that can lead to a reduction in effective surfactant concentration and alter rock wettability. In the present study, the process of surfactant adsorption/desorption on the rock surface was not accounted for; this aspect is planned for future investigation.

6. Conclusions

A systematic numerical study of oil displacement processes from terrigenous and cavernous-fractured media models using AES surfactant solutions has been conducted. The surfactant concentration varied from 0 to 0.1 wt.%. Preliminary laboratory studies of the properties were performed for the surfactant solutions under consideration. It was shown that increasing the AES concentration to 0.1 wt.% leads to a monotonic decrease in the solution’s interfacial tension by more than a factor of three compared to water. In contrast, the dependence of the contact angle on AES concentration is non-monotonic.

A numerical investigation of the oil displacement process from terrigenous and cavernous-fractured media models was carried out using AES surfactant solutions at various mass concentrations, incorporating their experimentally measured properties. The results demonstrate that for both types of reservoir models, the displacement coefficient (K_disp) increases significantly with rising surfactant concentration. It was established that at an AES concentration of 0.1 wt.%, K_disp increases by 15% for the terrigenous model and by 19% for the cavernous-fractured model compared to waterflooding.

It was shown that the most comprehensive criterion determining the increase in oil displacement coefficient from porous media and accounting for the influence of interfacial tension and wettability is the generalized capillary number ${Ca}^{\ast }$. For both reservoir models, K_disp increases monotonically with increasing ${Ca}^{\ast }$. It was found that at a fixed velocity of the displacing medium, the surfactant additive causes a local increase in the generalized capillary number by almost a factor of four, which is the primary mechanism for the K_disp enhancement in this case.