Simulation Study of CO₂ Huff-n-Puff in Tight Oil Reservoirs Considering Molecular Diffusion and Adsorption

Abstract

CO₂ injection has great potentials to improve the oil production for the fractured tight oil reservoirs. However, Current works mainly focus on its operation processes; full examination of CO₂ molecular diffusion and adsorption was still limited in the petroleum industry. To fill this gap, we proposed an efficient method to accurately and comprehensively evaluate the efficiency of CO₂-EOR process. We first calculated the confined fluid properties with the nanopore effects. Subsequently, a reservoir simulation model was built based on the experiment test of the Eagle Ford core sample. History matching was performed for the model validation. After that, we examined the effects of adsorption and molecular diffusion on the multi-well production with CO₂ injection. Results illustrate that in the CO₂-EOR process, the molecular diffusion has a positive impact on the oil production, while adsorption negatively impacts the well production, indicating that the mechanisms should be reasonably incorporated in the simulation analysis. Additionally, simulation results show that the mechanisms of molecular diffusion and adsorption make great contributions to the capacity of CO₂ storage in tight formations. This study provides a strong basis to reasonably forecast the long-term production during CO₂ Huff-n-Puff process.

1. Introduction

Unconventional reservoirs are always characterized by low permeability and porosity, which requires the techniques of hydraulic fracturing for the economic development [1,2,3,4]. Current studies show that the oil recovery after primary production is less than 10% [2], and tremendous amount of oil reserves in the tight formations. Hence, CO₂-EOR process has big potentials ₂-EOR process [5].

Recently, researchers have proved the great success of CO₂ injections in tight oil reservoirs theoretically and experimentally, which is beneficial to relieve global warming as well [6,7,8,9]. Due to the favorable injectivity and miscible displacement, CO₂-EOR performs better than water flooding at the pressure higher than MMP. CO₂-EOR includes two common scenarios: continuous CO₂ injection and CO₂ Huff-n-Puff process. In the huff-n-puff process, the production well is first transferred to the CO₂ injection well, stimulating the region near the fractures, after shut-in period, it is converted back to production. Zuloaga-Molero et al. [10] compared the efficiency of these two CO₂ injection scenarios in the Bakken formation, concluding that oil production is higher in CO₂ Huff-n-Puff process than CO₂ flooding in the low-permeability formation. Sanchez Rivera [11] investigated the main operation parameters during the CO₂ Huff-n-Puff process and evaluated their impacts on the CO₂-EOR performance. According to the work by Hawthorne et al. [6], the CO₂ injection in tight oil reservoirs is defined as the following conceptual steps: CO₂ first flows into and through the fractures; then it diffuses into the matrix and oil moves out of pores through swelling and reduced viscosity; finally, the oil will be driven into the fractures and the wellbore with the CO₂ pressure gradient. Recently, Janiga et al. [12] developed a data-driven model of huff-n-puff process using genetic programming. They optimized the huff-n-puff process with the laboratory data providing the good input data. Wan et al. [13] applied the NMR measurements to investigate the N₂ huff-n-puff efficiency at different cycles, illustrating NMR technique can well investigate the cyclic N₂ injection.

From the literature survey, the physical mechanisms i.e., nanopore effects, molecular diffusion, and adsorption influence the effectiveness of CO₂-EOR process. Hoteit [14,15] has pointed out that the mechanism of molecular diffusion plays a more important role in tight oil reservoirs than viscous forces and gravity drainage. Nanopores exist in the shale and tight formations; hence, high capillary pressure in nanopores will result in the unique fluid and interaction characteristics [16,17,18]. Additionally, Sapmanee [19] pointed out that fluid properties and flow conditions in confined pores should be modified due to pore proximity effects. Zhang et al. [20] modified van der Waals equation of state to investigate the CO₂ storage process in the fractured nanopores considering adsorption, concluding that phase behavior of CO₂ streams in fractured nanopores differs from bulk phases. Zhang et al. [21] applied the PR-EOS to evaluate solubility parameter in nanoscale and applied it to calculate the MMP in the CO₂ injection.

Kang et al. [22] studied CO₂ adsorption on two Barnett shale samples, pointing out that adsorption of CO₂ is 5–10 times greater than CH₄, releasing the adsorbed CH₄ in the matrix. Cipollar et al. [23] and Yang et al. [24] performed numerical studies to investigate the adsorption on the oil production. However, molecular diffusion was not included in their models. In the model built by Clarkson and Haghshenas [25] and Haghshenas et al. [26], they conducted the simulation of CO₂ Enhanced Recovery accounting for the fluid adsorption and diffusion, but the nanopore effects were neglected. Yu et al. [7] calculated the contributions of CO₂ molecular diffusion in improving the oil production in the Middle Bakken formations, concluding that large molecular diffusion leads to high incremental oil recovery. However, adsorption and nanopore effects were neglected in their study. In order to handle the adsorption of multi-components, Yang et al. [27] extended the Langmuir model and applied it to investigate primary production and CO₂-EOR process considering the gas adsorption. Results show that multi-component adsorption behavior slightly increases the oil recovery in the primary production but more complex in the CO₂-EOR process. Wan and Mu [28] concluded that the nanopore effect impacts the recovery of Eagle Ford shale gas condensate reservoir during the CO₂ cyclic injection. However, the molecular diffusion and adsorption were not considered. Zhang et al. [9] designed a sensitivity analysis of the effect of diffusion mechanism on the incremental oil recovery factor. They pointed out that the molecular diffusion has positive impacts on the performance of CO₂-EOR process in the tight oil reservoirs. However, their studies did not include the adsorption mechanism. Sun et al. [29] and Wang and Yu [30] used compositional model to investigate the CO₂-EOR with complex fracture geometries, but the nanopore effects and the adsorption were not considered. Czarnota et al. [31] studied the mass transfer between CO₂ and oil phase and applied the acoustic pulse-echo technique to accurately examine the oil volume expansion factor, which is important during the CO₂-EOR process.

In this work, we presented a useful approach to efficiently determine the effects of molecular diffusion and adsorption on the CO₂ Huff-n-Puff process effectiveness in the tight reservoirs. We first introduced the calculation of the fluid properties shifts considering the nanopore confinement. Then, A 3D reservoir model was built with the characterized properties from the Eagle Ford formation. Subsequently, history matching was conducted including the nanopore effects. After that, we performed the simulation studies of different CO₂-EOR process, including five cycles in 15 years. Each cycle includes one-month injecting, one-month soaking, followed by one year of back production. The well performance in different scenarios was compared to determine the contributions of the molecular diffusion and adsorption on the increment of oil production. CO₂ storage capacity considering the diffusion and adsorption was also analyzed. The simulation results indicate that the mechanisms of nanopore effect, molecular diffusion, and adsorption impact the well production in the tight oil reservoirs.

2. Methodology

2.1. Critical Property Shifts in Nanopores

As reported, the fluid PVT properties is significantly influenced for the pore size of 10 nm or less because of the surface-molecule interactions [32,33]. In this approach, only critical temperatures and pressures in nanopores are considered, which are calculated by the correlations published by Singh et al. [34], as presented in Equations (1)–(3).

$$\Delta T_c^{*}=\frac{T_{cb}-T_{cp}}{T_{cb}}=0.9409\frac{{\sigma }_{LJ}}{r_p}-0.2415{(\frac{{\sigma }_{LJ}}{r_p})}^2$$

(1)

$$\Delta P_c^{*}=\frac{P_{cb}-P_{cp}}{P_{cb}}=0.9409\frac{{\sigma }_{LJ}}{r_p}-0.2415{(\frac{{\sigma }_{LJ}}{r_p})}^2$$

(2)

$${\sigma }_{LJ}=0.244\sqrt[3]{\frac{T_{cb}}{P_{cb}}}$$

(3)

where r_p is the pore-throat radius. T_cb, T_cp, $\Delta T_c^{*}$ are the critical temperature in the bulk phase, critical temperature in the confined phase, and relative critical temperature shift, respectively. P_cb, P_cp, and $\Delta P_c^{*}$ are the critical pressure in the bulk phase, critical pressure in the confined phase, and relative critical pressure shift, respectively. ${\sigma }_{LJ}$ is the Lennard-Jones size parameter (collision diameter). Reid et al. [35] have summarized the critical properties of pure component in their work.

2.2. Flow Governing Equations

With the typical fluid and reservoir properties, a reservoir simulation model was proposed using CMG [36]. For component i, the total mass balance in different phases can be evaluated as follows [37]:

$$\frac{\partial }{\partial t}(\varphi {\displaystyle \sum _{l=1}^{N_p}{\rho }_l{S}_l}{w}_{il})+\overrightarrow{\nabla }\cdot ({\displaystyle \sum _{l=1}^{N_p}{\rho }_l}{w}_{il}{u}_l-\varphi {\rho }_l{S}_l{\overline{\overline{K}}}_{il}\nabla w_{il})-r_i=0,i=1,\dots ,N_p$$

(4)

where ${\rho }_l$ is density, w_il is mass fraction of component i. $S_l$ is the saturation. The subscript “l” represents the phase l. N_p is number of phases. u_l is velocity in Darcy’s flow, and $\varphi$ represents the porosity. r_i is mass rate as source or sink term in the injection or production.

${\overline{\overline{K}}}_{il}$ is the dispersivity coefficient of component i in the phase l, which is expressed as

$${\overline{\overline{K}}}_{il}=\frac{{\overline{\overline{D}}}_{il}}{\tau }+\frac{{\overline{\alpha }}_l\left|{\overline{u}}_l\right|}{\varphi S_l}$$

(5)

In Equation (5), we applied the Sigmund’s work to calculate the diffusion coefficient of component i (${\overline{\overline{D}}}_{il}$), which is valid for both oil and gas phases [38,39]. $\tau$ is tortuosity. ${\overline{\alpha }}_l$ is the dispersivity coefficient of phase l.

u_l is described as

$${\overline{u}}_{{}_l}=-\frac{\overline{\overline{k}}}{{\mu }_l}(\overline{\nabla }{p}_l-{\rho }_l\overline{g})$$

(6)

where p_l is pressure and ${\mu }_l$ is the viscosity. $\overline{\overline{k}}$ is permeability tensor. The subscript “l” represents the phase l. P_l is the pressure of phase l. The partial differential equations are solved by the CMG-GEM numerical reservoir simulator [36].

Yu and Sepehrnoori [40] have optimized the well spacing and pointed out that four horizontal wells are often drilled in the area of 1 mile × 2 miles. Luo et al. [41] found that the number of perforation stages is averagely 30 for each long lateral. Hence, we set a simulation model in the field scale, with the dimension of 10,560 ft, 5280 ft, and 40 ft in x, y, and z direction, respectively. The grid size is 40 × 40 × 40 ft. Four horizontal wells are zipper patterns with lateral length of 9280 ft and well spacing of 1360 ft. There are 30 hydraulic fractures along each well, as shown in Figure 1. The fracture spacing is 320 ft, and fracture height is assumed equal to the reservoir thickness. Parameters of reservoir and fracture are reasonably set for this field-scale case study, as summarized in

Table 1. Properties of reservoir and fracture from the Eagle Ford formation.

Parameters	Value	Unit
Initial reservoir pressure	7500	psi
Reservoir temperature	270	°F
Production time	15	years
Matrix porosity	12%	-
Total compressibility	1 × 10−6	psi−1
Matrix permeability	0.001	mD
Initial water saturation	17%	-
Fracture height	40	ft
Fracture spacing	320	ft
Fracture half-length (HM parameter)	220	ft
Fracture conductivity (HM parameter)	48	mD-ft

[42]. We used the local grid refinement (LGR) to describe the flow between the matrix and fractures. The grid of each block containing hydraulic fracture is set as 7 × 1 × 1. It should be noted that stress-sensitivity is neglected in this study.

Since the detailed fluid composition data for the Eagle Ford well is not available, we assumed six pseudo-components to represent the typical fluid in this study: CO₂, N₂, C₁, C₂-C₅, C₆-C₁₀, and C₁₁₊, respectively. Detailed composition and properties can be found in our previous work [43]. After tuning, oil gravity is 40 °API, GOR is 1900 scf/stb, and B_o is 1.65 rb/stb, which agree well with the properties measured by Orangi et al. [44]. We implement these properties into the PR-EOS [45]. Flash calculations are performed to determine the oil properties under the reservoir temperature of 270 °F. Details can be found in our previous work [46].

2.3. History Matching with An Actual Well from Eagle Ford Tight Oil Reservoir

In order to validate the reservoir simulation model, history matching was performed using the production data of a well from the Eagle Ford tight oil reservoir [47]. According to the experimental data on the Eagle Ford cores, the percentage of pores less than 20 nm takes around 80% in total [48]. Hence, the nanopore effects should be considered in the reservoir simulation. Sensitivity analysis of region numbers was also conducted, and division of four regions was finally decided: less than 5 nm, 5–10 nm, 10–20 nm, and larger than 20 nm, representing the pore size distribution of Eagle Ford formation. The contribution of each region is 42%, 27%, 13%, and 18% in total, respectively. Each gridblock has its own region and each region has its PVT properties. The values of critical properties for different pore sizes were calculated using Equations (1)–(3), as displayed in Figure 2.

The phase diagrams for different pore sizes are presented in Figure 3. As shown, the two-phase region becomes smaller and the bubble point pressure decreases at the temperature of 270 °F.

The nanopore effects on gas- oil ratio, formation volume factor, and viscosity are shown in Figure 4. With the suppression of pore size, viscosity decrease, formation volume factor and gas- oil ration will increase. It should be noted we applied the compositional reservoir simulator to investigate the CO₂ Huff-n-Puff process.

In order to implement the effect of pore size in the compositional simulation, we calculated the critical temperature and pressure of different components under different pore sizes using Equations (1)–(3). Then, keyword “EOSSET” was used to input the PVT properties and “EOSTYPE” was used to assign the regions for all grids in the CMG input file. After that, we run the CMG-GEM simulator to obtain the results of CO₂-EOR process.

In this section, we include the different pore size regions with different confined fluid properties to consider the nanopore effects on the history matching, which can obtain a more reasonable simulation model [49]. The diffusion and adsorption were not considered in the history matching process. During the history matching, the simulation constraint input is the oil flow rate, as shown in Figure 5A, while the water flow rate, gas flow rate, and BHP are considered as the history-matching parameters. The black circle represents the field data, while the red line is the simulation results. We tuned the parameters of fracture half-length, fracture conductivity, and relative permeability curves to obtain good results.

As presented in Figure 5B–D, the simulation results show great agreements with the actual production data. Additionally, based on the good agreements, the fracture conductivity and fracture half-length are reset as 48 mD-ft and 220 ft, respectively. Figure 6 presented the determined water-oil and liquid-gas relative permeability curves. The relative permeability curves remain the same in all grids in the following cases, and the non-Darcy effect is neglected.

3. Results and Discussion

In this study, the molecular diffusion and adsorption of CO₂ were both included in the reservoir simulation model. All reservoir properties keep the same as the reservoir model in Table 1 and history matching results. All models are assumed as no flow boundary condition. Mohebbinia et al. [50] stated that molecular diffusion transportation is important in tight rocks. According to the previous study, the range of measured CO₂ diffusion coefficients in Decane is about 1.97 × 10⁻⁵ to 1.26 × 10⁻⁴ cm²/s when the temperature is 100 °F and pressure is up to 850 psi [51]. Kumar and Mittal [52] presented that diffusion coefficient of CO₂ at the super-critical condition is much higher than that under the liquid condition. Since we have performed the sensitivity analysis of diffusion coefficients ranging from 0.0001 to 0.01 cm²/s [43], in this study, we set the CO₂ diffusion coefficients of 0.001 cm²/s in this study. Reasonable measurement of CO₂ diffusion coefficients should be conducted in the further study. The original data from Ambrose et al. [53] was applied to investigate the effect of adsorption. As shown in Figure 7, CO₂ has a larger adsorption than the light components (Methane and Ethane). Hence, the diffusion coefficients and adsorption of other components are equal to zero [54]. It should be noted that the adsorption in this study is assumed as single layer and only CO₂ adsorption is considered. The Langmuir model is applied to evaluate the mechanism of adsorption in this model.

We performed four cases to compare the effectiveness of CO₂-EOR process. Different mechanisms are shown in

Table 2. Physical mechanisms in different cases.

Case	CO2 Huff-n-Puff	Molecular Diffusion	Adsorption
Case A	N	N	N
Case B	Y	N	N
Case C	Y	Y	N
Case D	Y	Y	Y

.

Case A is set as the base case, in which primary depletion of the wells was investigated for 15 years. In the other three cases of CO₂ Huff-n-Puff process, the four horizontal wells experience five cycles. The wells experience three years of primary depletion at the beginning, then they are converted to injection wells and begin one-month CO₂ injection. Afterwards, the wells are shut in for one-month soaking. During this soaking time, CO₂ diffuses into the matrix and contact with oil. Finally, the horizontal wells are put back to production again. A new cycle will repeat after one-year production. The CO₂ injection rate of in the process is set as 5000 Mscf/day. Figure 8 illustrates the three stages of CO₂ Huff-n-Puff process: CO₂ injection, soaking period, and back production. According to the practical production history, BHP is constrained as 2000 psi in the Huff-n-Puff scenarios.

We evaluated the well performance at the end of production time, and the oil recovery factors of different scenarios are shown in Figure 9. Compared the base case (Case A) and Case B, the CO₂ EOR process contributes to the incremental oil recovery of 0.7%. Considering the CO₂ molecular diffusion (Case C), the oil recovery is 4.8% higher than that of the base case.

Figure 10 displays the oil viscosity and the swelling factor under different CO₂ concentrations. In this study, oil swelling factor is defined as “the volume of the oil after CO₂ injection divided by the volume of the oil before CO₂ injection into the cell” [55]. As shown, the oil viscosity decreases and the swelling factor with the increase of the CO₂ concentrations, indicating that the CO₂ injection is beneficial for the oil swelling and increment of oil production. Additionally, the oil viscosity and swelling factor will further decrease for smaller pore sizes.

Figure 11 displays the CO₂ distribution in Case B and Case C. As shown in Figure 11A, without considering the molecular diffusion, the CO₂ molecules just penetrate into the matrix near the hydraulic fractures instead of moving into the matrix. Therefore, CO₂ EOR process cannot be efficient as expected. When the molecular diffusion is considered, as shown in Figure 11B, it performs as the main driven force in the CO₂ EOR process, promoting more CO₂ molecules moving from hydraulic fractures to the matrix and mixing with the oil phase. CO₂ would swell more oil and reduce oil viscosity, leading to the larger amount of oil production. Therefore, molecular diffusion is a significant mechanism during CO₂ injection, which should be included in evaluating the performance of cyclic CO₂ injection.

As presented in Figure 9, the ultimate oil recovery in Case D, which considering molecular diffusion and adsorption performs 2.1% higher than the base case (Case A) with primary production at the end of simulation, but 2.7% lower than Case C (only includes molecular diffusion), indicating that adsorption has negative influence on the well performance. The reason may be that the adsorption potential of methane and ethane is lower than CO₂, resulting in desorption during the CO₂-EOR process. However, the heavy components are not desorbed even with the CO₂ injection. Hence, consideration of adsorption leads to less hydrocarbon production and smaller increase in the oil recovery.

The distribution of CO₂ mole fraction clearly illustrates the impact of adsorption mechanism. Compared Figure 12 with Figure 11B, CO₂ molecules do not move from fractures into matrix as far as the case only including the molecular diffusion (Case C). Less CO₂ molecules will move into the oil phase, resulting in the weak oil mobility. Additionally, adsorption would decrease the injected volume of CO₂ diffusing into shale oil because some portion of injected CO₂ has been adsorbed into the surface. Therefore, the adsorption performs a negative effect on the CO₂-EOR process.

The impacts of molecular diffusion and adsorption on the cumulative CO₂ backflow was compared as shown in Figure 13. The black line represents the total CO₂ injection volume. According to the simulation results in Figure 13, around 79% of injected CO₂ will be produced back at the end of simulation time without molecular diffusion and adsorption (Case B). While about 67% of injected CO₂ will flow back after 15 years only with molecular diffusion (Case C). Hence, molecular diffusion leads to large amount of CO₂ trapping in the tight formations. Additionally, a better mixture of CO₂ molecules and oil phase generates, resulting in higher cumulative oil production. If adsorption is considered (Case D), the cumulative CO₂ backflow decreases to 59%. The mechanism of adsorption is positive to the CO₂ storage, leading to more CO₂ molecules trapped and less backflow to the ground after soaking. Since no CO₂ molecules are injected in the primary production case (Case A), the percentage of CO₂ backflow and relative CO₂ storage are 0.

The difference of the average reservoir pressure is depicted in Figure 14. As shown, the average reservoir pressure of the base case with the primary production is the lowest. When the molecular diffusion and adsorption are included, the average reservoir pressure increases. The pressure of the case C with only molecular diffusion is higher, leading to the largest increment of oil production.

Figure 15 presented the pressure distribution after 15-year production. As shown, the pressure of the grid near the hydraulic fracture drops quickly. The pressure in Case C (Figure 15D) is higher than other cases. Molecular diffusion accelerates the pressurized CO₂ molecules to diffuse from the fractures to the tight formations and light and intermediate hydrocarbons are extracted from the oil. Considering adsorption would decrease the injected volume of CO₂ diffusing into the tight oil; therefore, the pressure in Case D is a little higher than Case C, as displayed in Figure 15C,D. It also implies that neglecting the molecular diffusion and adsorption will underestimate the well performance in the tight oil reservoirs. Therefore, these mechanisms should be reasonably included in the investigation of CO₂-EOR process.

4. Conclusions

In this study, we proposed a useful compositional approach to simulate the CO₂-EOR process in the Eagle Ford formation. Physical mechanisms of molecular diffusion and adsorption were accurately examined in this typical model. The conclusions from this study are obtained as follows:

(1)

Simulation results illustrate that CO₂ molecules move from fractures to matrix with the mechanism of CO₂ molecular diffusion, which can efficiently improve the oil recovery factor in the CO₂-EOR process. The oil recovery increases by 4.8% with the diffusion coefficient of 0.001 cm²/s compared to the base case.

(2)

Compared to the case with molecular diffusion, the mechanism of adsorption influences CO₂ penetration into the matrix, performing a negative influence on the incremental oil recovery of the CO₂ Huff-n-Puff process. In this study, the increment of oil recovery is 2.1% considering both molecular diffusion and adsorption, 2.7% lower than the case only including the molecular diffusion.

(3)

Results indicate that molecular diffusion and adsorption will decrease the cumulative CO₂ backflow in the simulation of CO₂-EOR process. For this typical reservoir, the cumulative CO₂ backflow decreases to 59%. Hence, these mechanisms should be properly considered in the analysis.