Introduction

It is widely accepted that carbon dioxide (CO2) displacement is an effective and promising method to enhance tight oil recovery (Torabi et al. 2012; Yu et al. 2020; Chen et al. 2020; Song and Yang 2017; Assef et al. 2019; Hartono et al. 2023). It is also environmentally friendly because it can reduce CO2 emission by carbon capture and storage (CCS) (Yu et al. 2015; Rezvani1 and Rafiei 2023). CO2 displacement can be divided into immiscible displacement, near miscible displacement and miscible displacement. The most of tight oil reservoirs pressure is lower than minimum miscibility pressure (MMP) in China. Dong et al (2000) found that CO2 near miscible displacement has a high percentage. Therefore, CO2 near miscibility study is significant.

There are many existing studies on CO2 near miscible displacement. Stalkup (1978), Zick (1986) and Novosad (1991) analyzed a large number of field miscible displacement based on slim tube experiments and state equation calculations. They found that a non-traditional miscible displacement was possible to exist. Orr and Silva (1982), Orr and Jensen (1984) found that there was a transition pressure range from immiscibility to miscibility in recovery curves of slim tube experimental results. Oil and CO2 were in a state of near miscible in the pressure range. Zick (1986) first proposed the near miscible displacement concept. Near miscible displacement is both affected by condensate and evaporation. It is characterized by a low interfacial tension (IFT) between oil and gas. Near miscible displacement recovery is more than 95%. However, IFT of near miscible displacement is not quantified. Thomas et al. (1994) systematically analyzed a large number of miscible displacement field data. They found that many successful and effective miscible displacement projects were not completely miscible around the world. Then, they proposed that it is more practical and economic to improve CO2 near miscible displacement performance than CO2 miscible displacement performance. Shyeh-Yung and Stadler (1995), Grigg et al. (1997), Schechter et al. (1998), Dong et al (2000) successively confirmed the existence of a special pressure range near MMP based on long core displacement experiments. The recovery efficiency in this pressure range did not decrease significantly with the decrease in CO2 injection pressure. Bui et al. (2010) proposed that the pressure interval of CO2 near miscible displacement in Ogallah oilfield was from 0.8 to 1 times MMP based on their experience. Yang (2011) proposed that CO2 near miscible displacement appeared when the IFT was 0.5 mN/m according to the slim tube experiments. However, there is no existing method to study the effect of near miscible region on oil recovery.

Therefore, this paper aims to propose an approach to calculate oil recovery considering near miscibility. Slim tube experiments are conducted to study near miscible pressure range. A three-stage equation is built to characterize the correlation between oil recovery and pressure. Pressure distribution is discretized between CO2 injection well and oil production well according to threshold pressure gradient of tight oil reservoirs. Then, a new approach is presented to calculate oil recovery. This approach takes near miscibility into account. At last, the effect of near miscibility on CO2 displacement recovery and optimal pressure level is analyzed.

Slim tube experiments

Experimental apparatus

Slim tube experiments are a standard method to measure CO2 MMP in petroleum industry. They can simulate actual development processes. They are regarded as the most accurate way to determine MMP. In this study, slim tube experiments were conducted to study near miscible characteristics. The slim tube systems include a high-pressure positive displacement syringe pump, a floating piston high-pressure accumulator for oil, a floating piston high-pressure accumulator for gas, a thermostat, a slim tube, a back-pressure regulator, liquid meter and gas meter (Fig. 1). The packing material of the slim tube is quartz sand. Table 1 is the basic parameters of slim tube.

Fig. 1
figure 1

The sketch of the apparatus used in the slim tube experiment

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Table 1 The basic parameters of slim tube
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Experimental materials

The crude oil comes from BZ25-1 oilfield. BZ25-1 oilfield is located in Bohai bay basin. It is a tight oil reservoir. Its depth is from − 3500 to − 3900 m. The average porosity is 13% and the average permeability is from 5 to 10 mD. The reservoir temperature is 150℃ and the original reservoir pressure is 55 MPa. The density of degassed oil is 0.8685 g/cm3 when pressure is 0.101 MPa and temperature is 20 ℃. Solution gas oil ratio (GOR) at reservoir condition is 89 m3/m3. The oil saturation pressure is 16 MPa. The oil viscosity at reservoir condition is 1.13 mPa·s. Table 2 is the basic parameters of BZ25-1 oilfield. The crude oil samples' components were analyzed using JEFRI PVT analyzer manufactured by Canadian DBR company. Table 3 shows the components of the crude oil samples. The injection gas used in the experiment is ultra-high pure CO2 (99.9%).

Table 2 The basic parameters of BZ25-1 oilfield
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Table 3 The components of BZ25-1 degassed oil
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Experimental procedures

Kerosene was first injected at the experimental temperature and pressure to clean the slim tube. Then, the crude oil was used to displace the kerosene in the slim tube at a rate of 60 cm3/h. The properties of the produced fluids were measured every 0.1 times pore volume (PV) when the injection volume was 2 times PV. The properties included components, viscosity, density, GOR. Oil displace stopped and the slim tube was ready if the properties of the produced fluids were the same as those of the oil sample. The injected CO2 gas was stored in the floating piston high-pressure accumulator at the experimental temperature. The CO2 gas was injected at a rate of 6 cm3/h into the slim tube. The back-pressure regulator was used to control the slim tube pressure. The slim tube experiments need to be added near MMP in order to study near miscibility. The produced fluids went across the back-pressure regulator and were separated into the gas meter and liquid meter. The metering system recorded the experiment data including density, viscosity and volume of the produced gas and oil. The total volume of the injected gas was 1.2 PV.

Approach to calculate recovery of CO2 displacement

Correlation between oil displacement efficiency and pressure considering near miscibility

The relationship of oil displacement efficiency and pressure can be obtained from the slim tube experiment results (Fig. 2). It can be divided into 3 parts according to the slope difference. The linear function is the most optimal based on the relevance of different functions (Table 4). Then, the correlation of oil displacement efficiency and pressure is established (Eqs. 1–3). The six constants (ai, bi, an, bn, am, bm) in 3 parts are obtained by regression the slim tube experiment results from Fig. 2 (Table 5). The critical pressure pin is obtained when Eq. 1 equals Eq. 2. The other critical pressure pnm is obtained when Eq. 2 equals Eq. 3.

$$E_{\rm o} = {a_{\rm i}}p + {b_{\rm i}},\;p < p_{\rm in}$$
(1)
$$E_{\rm o} = {a_{\rm n}}p + {b_{\rm n}},\;{p_{\rm in}} < p < {p_{\rm nm}}$$
(2)
$$E_{\rm o} = {a_{\rm m}}p + {b_{\rm m}},\;p > {p_{\rm nm}}$$
(3)

where Eo is oil displacement efficiency, %; p is pressure, MPa; ai, bi is constant when it is immiscible, dimensionless; an, bn is constant when it is near miscible, dimensionless; am, bm is constant when it is miscible, dimensionless; pin is critical pressure between immiscibility and near miscibility, MPa; pnm is critical pressure between near miscibility and miscibility, MPa.

Fig. 2
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The oil displacement efficiency results of the slim tube experiment

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Table 4 The relevance of different regression functions
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Table 5 The constants in Eqs. 1, 2, 3
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The oil displacement efficiency increases with the increase in pressure. The oil displacement efficiency increases fastest, and the slope of the oil displacement efficiency and pressure curve is maximal when it is immiscible. The component exchange between CO2 and crude oil begins to intensify with the increase in pressure. An inflection point appears in the IFT curve when pressure increases to reach the critical pressure pin (Fig. 3). The change of displacement mechanism and IFT curve leads to the slope difference of oil displacement efficiency. The slope of the oil displacement efficiency and pressure curve decreases to be 1.135 when it is near miscible. Then, IFT decreases to close to 0 and oil displacement efficiency is close to 90% when pressure increases to reach the critical pressure pnm (Fig. 3). The oil displacement efficiency does not change significantly with the increase in pressure when the oil displacement efficiency exceeds 90%. It is completely miscible. The slope of the oil displacement efficiency and pressure curve further decreases to 0.3425.

Fig. 3
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The IFT results of the slim tube experiment

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Pressure distribution in tight oil reservoirs

Nonlinear flow in porous media is obvious in tight oil reservoirs. Threshold pressure gradient is an important parameter to characterize the nonlinear flow in porous media. It affects pressure between injection wells and production wells in tight oil reservoirs. However, the threshold pressure gradient of CO2 and oil is different (Fig. 4). CO2 is dissolved in crude oil when it is injected into oil reservoirs. The density and viscosity of crude oil dissolving CO2 decrease. As a result, the threshold pressure gradient of CO2 is only 0.3 times that of oil. There is an additional pressure drop considering threshold pressure gradient. Therefore, the pressure distribution of between injection wells and production wells in tight oil reservoirs can be expressed in Eq. 4 (Tian et al. 2018). Formation pressure increases because of CO2 injection. CO2 injection well bottom hole pressure affects pressure distribution when formation pressure is higher than original formation pressure. The pwf is injection well bottom hole pressure when formation pressure is higher than original formation pressure. The pwf is production well bottom hole pressure when formation pressure is lower than original formation pressure. It is found from Fig. 5 that the pressure is mainly consumed in where the near-well region due to the greater flow resistance and the smaller flow area.

$${p_r} = {p_{wf}} + \frac{{\left( {{p_{e - }}{p_{wf}}} \right) - G\left( {{r_e} - {r_w}} \right)}}{{ln\frac{{r_e}}{{r_w}}}} \cdot ln\frac{r}{{r_w}} + G\left( {r - {r_w}} \right)$$
(4)

where pr is the pressure when the distance to production well is r if pr < pe, else pr is the pressure when the distance to CO2 injection well is r, MPa; pwf is the bottom pressure of production well if pr < pe, else pwf is the bottom pressure of gas injection well, MPa; pe is original formation pressure, MPa; re is the distance to oil well where the pressure is pe, m; rw is well radius, m; G is threshold pressure gradient of CO2 displacement oil, MPa/m; r is the distance to production well if pr < pe, else r is the distance to CO2 injection well, m.

Fig. 4
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Threshold pressure gradient of oil and CO2 displacement oil

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Fig. 5
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Pressure distribution in tight oil reservoirs

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Discrete well space to calculate oil recovery

The pressure from CO2 injection well to production well is different during the development of CO2 displacement. Pressure determines oil recovery efficiency and miscible degree. As a result, oil displacement efficiency from CO2 injection well to production well is also different. The space between CO2 injection well and production well are divided into many micro-units (Fig. 5). It is assumed that the pressure in each micro-unit cell is constant. The oil recovery efficiency and miscible degree in each micro-unit cell is constant as well. The cumulative oil production of CO2 displacement in the micro-unit cell at the distance r from the production well can be expressed to be Eq. 5. The cumulative oil production expression of injection and production well group is yielded by integrating Eq. 5 with respect to distance (Eq. 6). Oil displacement recovery is obtained (Eq. 7) when Eq. 6 is divided by oil reserves when the distance between oil well and injection well is d. Equations 1–4 and Eq. 6 were substituted into Eq. 7 to obtain the oil recovery considering the near miscibility.

$$d{Q_o} = 2\pi r \cdot dr \cdot h \cdot \emptyset \cdot {S_o} \cdot {E_o} \cdot \alpha$$
(5)
$${Q_o} = \mathop \sum \limits_{r_w}^d 2\pi r \cdot h \cdot \emptyset \cdot {S_o} \cdot {E_o} \cdot \alpha \cdot dr = \mathop \int \limits_{r_w}^d 2\pi r \cdot h \cdot \emptyset \cdot {S_o} \cdot {E_o} \cdot \alpha dr$$
(6)
$${R_o} = \frac{{Q_o}}{{N_o}} = \frac{{2\pi \cdot h \cdot \emptyset \cdot {S_o} \cdot \alpha }}{{N_o}}\mathop \int \limits_{r_w}^d r\left\{ {a\left[ {{p_{wf}} + \frac{{\left( {{p_{e - }}{p_{wf}}} \right) - G\left( {{r_e} - {r_w}} \right)}}{{ln\frac{{r_e}}{{r_w}}}} \cdot ln\frac{r}{{r_w}} + G\left( {r - {r_w}} \right)} \right] + b} \right\}dr$$
(7)

where dQo is the cumulative oil production where the distance to oil well is from r to r + dr in CO2 displacement, m3; Qo is the cumulative oil production where the distance to oil well is r in CO2 displacement, m3; Φ is porosity, dimensionless; So is oil saturation, dimensionless; d is the distance between oil well and injection well, m; dr is the distance of micro-units between CO2 injection well and production well, m; α is sweep efficiency, dimensionless; Ro is CO2 displacement oil recovery when the distance between oil well and injection well is d, dimensionless; No is oil reserves when the distance between oil well and injection well is d, m3.

Results and discussion

Effect of near miscibility on minimum miscibility pressure

Minimum miscibility pressure is the key parameter in the CO2 injection development. It is important to judge whether CO2 miscible displacement can be carried out. Existing method to calculate MMP is 2-stage method. The pressure is regarded to be MMP at the intersection of immiscible trend line and miscible trend line according to slim tube experiment results (Fig. 6). The 2-stage method ignores the near miscible region. Therefore, the comparison and analysis of MMP are conducted based on BZ25-1 oilfield.

Fig. 6
figure 6

Oil displacement efficiency versus pressure of 2-stage and 3-stage

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Figure 6 is the BZ25-1 oilfield slim tube experiment results. Table 2 shows the basic parameters of BZ25-1 oilfield. Table 3 shows the basic parameters of BZ25-1 oil. MMP of BZ25-1 is 36 MPa using 2-stage method, while MMP of BZ25-1 is 40 MPa using 3-stage method. MMP of 2-stage method is 4 MPa lower than that of 3-stage method. Minimum near miscible pressure (MNMP) is 31 MPa using 3-stage method. The near miscible pressure range is from 0.87 times 2-stage MMP to 1.12 times 2-stage MMP. It is also from 0.77 times 3-stage MMP to 1 time 3-stage MMP. Then, Fig. 7 is obtained when oil displacement efficiency from 2-stage method minus 3-stage method (Eq. 8). From Fig. 7, it is found that oil displacement efficiency difference of 2-stage method and 3-stage method first becomes larger and then decreases with increase in pressure. It reaches to be the maximum when the pressure is 36 MPa which is the MMP using 2-stage method. The maximum of the oil displacement efficiency difference is 3.4%.

$$\Delta E_{o} = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {p\left\langle {31\;{\text{MPa}}\;{\text{or}}\;p} \right\rangle 40\;{\text{MPa}}} \hfill \\ {\left( {1.9075p + 24.137} \right) - \left( {1.135p + 48.088} \right),} \hfill & {~31\;{\text{MPa}} \le p \le 35.7\;{\text{MPa}}} \hfill \\ {\left( {0.3425p + 80.023} \right) - \left( {1.135p + 48.088} \right),} \hfill & {~35.7\;{\text{MPa}} \le p \le 40\;{\text{MPa}}} \hfill \\ \end{array} } \right. = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {p\left\langle {31\;{\text{MPa}}\;{\text{or}}\;~p} \right\rangle 40\;{\text{MPa}}} \hfill \\ {0.7725p - 23.951,~} \hfill & {31\;{\text{MPa}} \le p \le 35.7\;{\text{MPa}}} \hfill \\ { - 0.7925p + 31.935,} \hfill & {~35.7\;{\text{MPa}} \le p \le 40\;{\text{MPa}}} \hfill \\ \end{array} } \right.$$
(8)

where ΔEo is oil displacement efficiency difference of 2-stage method and 3-stage method, %.

Fig. 7
figure 7

Oil displacement efficiency difference of 2-stage and 3-stage in near miscible region

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Effect of near miscibility on optimal formation pressure level

Although crude oil composition, reservoir temperature and other factors affect MMP, MMP of a specific reservoir is basically fixed. Thus, formation pressure level is the most important parameter to be optimized in a CO2 injection development design. It is key to miscible degree and development performance for a specific reservoir. Therefore, oil recovery is compared and analyzed for different formation pressures / MMP based on BZ25-1 oilfield. Table 2 shows the basic parameters of BZ25-1 oilfield. Table 6 shows the basic parameters of BZ25-1 oilfield CO2 injection development design. Oil recovery first appeared to increase rapidly with the increase in formation pressure/MMP. Then, the increase rate of oil recovery reduces. There is an inflection point. And the inflection point is the optimal formation pressure level. The optimal formation pressure level is 38 MPa when using 2-stage method without considering near miscibility (Fig. 8). It is the 1.06 times MMP from 2-stage method while 0.87 times from 3-stage method. This indicates that the optimal formation pressure level using 2-stage method is in near miscible region and it actually is not a miscible displacement. However, the optimal formation pressure level is 43 MPa when using 3-stage method with considering near miscibility. It is the 1.21 times MMP from 2-stage method while 1.08 times from 3-stage method. The optimal formation pressure using 3-stage method is 5 MPa larger than that using 2-stage method. Figure 9 shows the oil recovery difference between 2-stage method and 3-stage method. It is found that the oil recovery difference first becomes larger and then decreases as formation pressure increases. This indicates that the proportion of near miscibility area in the formation increases, and the effect on recovery efficiency increases. As the formation pressure continues to increase, the proportion of near miscibility area in the formation decreases, and the proportion of miscibility area increases, so the difference in recovery efficiency decreases. The oil recovery considering near miscibility is from 0.9 to 1.3% lower than that not considering near miscibility. It reaches to be the maximum when the pressure is 37.8 MPa which is the optimal formation pressure level using 2-stage method. The maximum of the oil recovery difference is 1.3%.

Table 6 The basic parameters of BZ25-1 oilfield CO2 injection development design
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Fig. 8
figure 8

The comparison of optimal formation pressure level using 2-stage method and that of using 3-stage method

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Fig. 9
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The recovery difference of 2-stage method and 3-stage method

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Conclusions

The important conclusions of this study can be summarized as follows:

  1. 1.

    A new approach is established to calculate oil recovery of CO2 displacement in tight oil reservoirs in this paper. The oil displacement efficiency curve is divided into immiscibility, near miscibility and miscibility. And three linear function is obtained, respectively. Then, the space between production well and CO2 injection well is discrete to characterize the effect of near miscibility on oil recovery.

  2. 2.

    Three-stage method is used to obtain the minimum miscibility pressure (MMP) and minimum near miscible pressure (MNMP) according to slim tube experiment results. This method considers the effect of near miscibility on oil displacement efficiency. Minimum miscibility pressure without considering near miscibility is 4 MPa lower than considering near miscibility. The near miscible pressure range is from 0.77 times to 1 time MMP considering near miscibility. Oil displacement efficiency difference reaches to be the maximum when the pressure is the MMP without considering near miscibility. The maximum of the oil displacement efficiency difference is 3.4%.

  3. 3.

    The optimal formation pressure considering near miscibility is 5 MPa larger than that without considering near miscibility. The oil recovery considering near miscibility is from 0.9% to 1.3% lower than that not considering near miscibility. It reaches to be the maximum when the pressure is the optimal formation pressure level without considering near miscibility. The maximum of the oil recovery difference is 1.3%.