A Multiphase and Multicomponent Model and Numerical Simulation Technology for CO₂ Flooding and Storage

Abstract

In recent years, CO₂ flooding has become an important technical measure for oil and gas field enterprises to further improve oil and gas recovery and achieve the goal of “dual carbon”. It is also one of the concrete application forms of CCUS. Numerical simulation based on CO₂-EOR plays an indispensable role in the study of the mechanism of CO₂ flooding and buried percolation, allowing for technical indicators to be selected and EOR/EGR prediction to be improved for reservoir engineers. This paper discusses the numerical simulation techniques related to CO₂ flooding and storage, including mathematical models and solving algorithms. A multiphase and multicomponent mathematical model is developed to describe the flow mechanism of hydrocarbon components–CO₂–water underground and to simulate the phase diagram of the components. The two-phase P-T flash calculation with SSI (+DEM) and the Newton method is adopted to obtain the gas–liquid phase equilibrium parameters. The extreme value judgment of the TPD function is used to form the phase stability test and miscibility identification model. A tailor-made multistage preconditioner is built to solve the linear equation of the strong-coupled, multiphase, multicomponent reservoir simulation, which includes the variables of pressure, saturation, and composition. The multistage preconditioner improves the computational efficiency significantly. A numerical simulation of CO₂ injection in a carbonate reservoir in the Middle East shows that it is effective for researching the recovery factor and storage quantity of CO₂ flooding based on the above numerical simulation techniques.

1. Introduction

As the global climate gradually warms, the greenhouse gas CO₂ poses a significant threat to human survival and social economic development. At present, CO₂ injection into oil and gas fields is one of the effective means of utilizing CO₂. On the basis of carbon reduction, this approach makes full use of the advantage of CO₂ flooding to improve the oil–gas recovery ratio. To date, CO₂-EOR technology has been widely applied in the United States, former Soviet Union, Canada, Great Britain, and other countries. Several foreign oilfields have effectively conducted field tests for CO₂ flooding. Some experimental studies on CO₂-EOR have been conducted since the 1960s in China. Field tests were first conducted in the Daqing oilfield in the 1980s, and then pilot tests were carried out in oil fields such as Daqing, Jilin, Shengli, and Changqing in the following decades. It has been proven that CO₂ flooding, which can increase the recovery rate by 40% or more, is one of the most promising EOR methods [1,2,3,4,5,6,7,8].

Since the temperature and pressure of most reservoirs are above the critical point of CO₂, the CO₂ injected under reservoir conditions is a supercritical fluid, which is a high-density gas. In terms of the thermal–physical properties, it has the dual characteristics of being a gas and a liquid; i.e., the density is higher than that of conventional gas and is close to a liquid, and it has the strength of a conventional liquid solvent. The viscosity is similar to that of gas, which is much less than that of liquid. The diffusion coefficient is also close to that of gas, which is about 10~100 times of that of liquid, so it has good fluidity [9,10].

The above-mentioned properties of CO₂ in the supercritical state are very important to improve oil and gas recovery. When a significant quantity of CO₂ is injected into the reservoir at high pressure, CO₂ will be dissolved in the crude oil because the reservoir temperature is significantly higher than the critical temperature. This leads to oil volume expansion and a reduction in the viscosity of the crude oil as well as a decrease in the interfacial tension between oil and water. Consequently, this process enhances the recovery rate of crude oil.

Due to the physical and chemical properties of CO₂, which is easily soluble in oil [9,10], and its supercritical physical properties, the displacement mechanism is mainly observable through the following aspects:

Reduced interfacial tension and displacement resistance. Due to the presence of a significant amount of CO₂ mixed with light hydrocarbons during the oil displacement process, the interfacial tension between oil and water is significantly minimized, resulting in reduced residual oil saturation and, ultimately, enhancing oil recovery.

Decreased viscosity of crude oil. The viscosity of crude oil can decrease significantly as CO₂ is dissolved with the extent of the decrease being influenced by factors such as pressure, temperature, and the original viscosity of the non-carbonated crude oil.

Expanded volume of crude oil. When carbon dioxide is completely dissolved in crude oil, the crude oil’s volume expands by 10% to 40%, boosting its kinetic energy and significantly reducing capillary and flow resistance during transportation. This ultimately enhances the crude oil’s flow capacity.

Extraction and gasification of light hydrocarbons from crude oil. During CO₂ displacement and soaking, the light components in crude oil can be gasified or extracted when the pressure exceeds a certain value. In particular, part of the residual oil that has not been released from the formation water after expansion is subject to interphase mass transfer with a CO₂ gas phase. The light components of the bound residual oil and CO₂ gas form a CO₂-rich gas phase, which is produced during CO₂ huff and puff, thus increasing the production of a single well.

Miscible flooding. The miscibility effect results in the extraction and gasification of light hydrocarbons in crude oil when mixed with CO₂, leading to the formation of a zone where CO₂ and light hydrocarbons are miscible. The miscible zone movement process is highly effective for displacing oil, resulting in an oil recovery rate exceeding 90% as the theoretical value.

Furthermore, CO₂ flooding can enhance the fluidity ratio of crude oil compared to water, increase the injection capacity through acidizing and unblocking, and facilitate gas dissolution during pressure drop stages. The above-mentioned mechanisms exist simultaneously in the process of CO₂-injection oil recovery, but the effect of each mechanism is different, which is determined by the reservoir lithology, fluid property, and development mode.

At the same time, some of the CO₂ is buried underground due to structural trapping, capillary binding, water dissolution formation, and rock reaction mineralization, which results in carbon reduction.

Currently seen as a crucial method for enhancing oil recovery, CO₂ flooding is increasingly being utilized and is gaining popularity due to its broad applicability, cost-effectiveness, and substantial enhancement of oil recovery. CO₂ flooding numerical simulation has also become a popular research topic. It is an important tool for the field selection of CCUS blocks, the calculation of technical indexes, and the prediction of oil and gas recovery (EOR/EGR) indexes. At the same time, because of the special properties of CO₂ gas and its special mechanism of seepage, the development of a mathematical model description of the CO₂ displacement process and the solution of EOS have become especially difficult.

Although some commercial software currently implements numerical simulation functions for CO₂ flooding and storage, there is no complete description of numerical simulation techniques in the literature, including mathematical models for seepage, P-T flash evaporation calculations, and linear-equation-solving algorithms. This study establishes a complete mathematical model for multiphase and multicomponent systems that can comprehensively describe the underground flow mechanism of hydrocarbon components–CO₂–water. Solving the extremum of the TPD function is adopted to determine whether the reservoir fluid is in a single-phase, two-phase, or mixed-phase state. On the basis of traditional SSI and Newton’s methods, the DEM method is used to calculate the critical points for two-phase P-T flash evaporation to converge faster and more accurately. A tailor-made multistage preconditioner is established for the linear equations of strongly coupled, multiphase, and multicomponent reservoir simulation, which is used to solve variables such as pressure, saturation, and composition. This algorithm can save a significant amount of time in numerical simulation calculations.

2. Key Numerical Simulation Techniques

2.1. Mathematical Model of CO₂ Flooding

In comparison to the traditional oil recovery process, the displacement mechanism and percolation law of CO₂ flooding are more complex, leading to more frequent changes in the phase and composition of subsurface fluids. The conventional black oil model is insufficient to fulfill the demands of CO₂ flooding numerical simulation. The commonly used mathematical models for CO₂ flooding include an improved black oil model, transport diffusion model (solvent model), near-component model (gas–liquid equilibrium constant model), and full component model (flash calculation model) [11]. In this study, an integrated multicomponent mathematical model is established to describe the variation process of complex hydrocarbon components containing CO₂.

The model incorporates the assumption that the flow behavior of oil–gas–water three-phase fluid adheres to Darcy’s law. It also considers the instantaneous completion of mass transfer, phase state changes, and phase equilibrium for each component. The water component maintains an independent phase state and does not partake in the interphase mass transfer of oil–gas, especially with the virtual component $N_c$. Additionally, the model accounts for rock compressibility and anisotropy, gravitational and capillary forces, and the impact of oil–gas interface tension on relative permeability. The seepage process is assumed to be isothermal flow [12,13].

For any component $i$, the flow in porous medium meets with the principles of mass and momentum conservation.

$$\nabla \cdot \left[{\displaystyle \sum _{j=1}^{N_p}\frac{K{k}_{rj}}{{\mu }_j}{\rho }_j{x}_{ij}\left(\nabla P+\nabla P_{c,j}-{\rho }_{j}g\nabla D\right)}\right]+q_i={\displaystyle \sum _{j=1}^{N_p}\frac{\partial }{\partial t}}\left(\varphi {\rho }_j{x}_{ij}{S}_j\right)$$

(1)

Due to the complex mass transfer in multiphase multicomponent systems, the distribution of any component among the phases follows the phase equilibrium principle.

The phase equilibrium equation can be defined by the principle of fugacity balance. For the multiphase phase equilibrium system:

$$f_{i,j}-f_{i,k}=0,\hspace{1em}i=1,2,\cdots ,N_c;\hspace{1em}j,k=1,2,\cdots ,N_p,j\ne k$$

(2)

In order to address the mathematical equations stated above, it is essential to integrate auxiliary equations such as the capillary force equation, saturation equation, and component normalization equation. The specific form is as follows:

Capillary pressure equation:

$$P_{cow}(S_w)=p_o-p_w$$

(3)

$$P_{cgo}(S_g)=p_g-p_o$$

(4)

Saturation equation:

$${\displaystyle \sum _{j=1}^{N_p}{S}_j=1}$$

(5)

Component normalization equation:

$${\displaystyle \sum _{i=1}^{N_c}{x}_{ij}=1\hspace{1em}},\hspace{1em}j=1,2,\cdots ,N_p$$

(6)

2.2. EOS of CO₂ Hydrocarbons

When CO₂ is injected into a reservoir, the resulting multicomponent hydrocarbon mixture may exhibit different phase states, such as gas, liquid, or a coexistence of the two states, under different temperature and pressure conditions. The phase state depends on the composition of the components. When the temperature and pressure of the system are constant, the phase state and composition of each phase in the system also stabilize, i.e., an equilibrium state. Therefore, to calculate the physical parameters of multicomponent mixtures containing CO₂ under different pressure conditions, it is necessary to first determine the phase state of the fluid, judge whether the mixture is gaseous, liquid, or two coexisting phases, and then determine the mole fraction of each phase and the distribution of each component. This is the calculation of the gas–liquid equilibrium process [14,15,16,17,18,19,20,21,22].

To calculate the volumes of the gas and liquid phases in hydrocarbon mixtures with CO₂, the composition of both the liquid and gas phases is necessary for prediction. In the process of CO₂ injection numerical simulation, the original fugacity equilibrium system is broken after updating a Newtonian step. Therefore, these components are unknown, regardless of whether there is a gas phase or a liquid phase. Therefore, the phase equilibrium calculation is carried out first to obtain the equilibrium gas- and liquid-phase components before passing through the state. The equation of state is used to determine the compressibility factors of the gas and liquid phases. Hence, calculating the phase equilibrium for the CO₂ injection process is incredibly important.

The phase equilibrium calculations discussed in this paper use the PR (Peng–Robinson) or SRK (Soave–Redlich–Kwong) equations of state with volume correction to calculate the compressibility factors (or molar volumes) of each component, and we also calculated the physical properties of hydrocarbon mixtures, including fugacity, Helmholtz free energy, density, viscosity, and interfacial tension. These are all cubic equations of state with three parameters. The general formula can be written as

$$Z^3+s{Z}^2+qZ+r=0$$

(7)

The equation for the three coefficients, where $Z$ represents the compressibility factor, is given as follows:

$$\left\{\begin{array}{l}s=\left(u-1\right)B-1\\ q=A+\left(w-u\right)B^2-uB\\ r=-AB-w{B}^2-w{B}^3\end{array}\right.$$

(8)

of which $A=\frac{ap}{R^2{T}^2},B=\frac{bP}{RT}$, and R is the gas constant. When calculating $Z$ of the mixture, $A$ and $B$ need to be calculated through the mixing law, that is, linear mixing laws for hydrocarbon mixture, namely,

$$\left\{\begin{array}{l}a={\displaystyle \sum _{i=1}^{Nc}{\displaystyle \sum _{j=1}^{Nc}{x}_i{x}_j\left(1-k_{ij}\right)\sqrt{a_i}\sqrt{a_j}}}\\ b={\displaystyle \sum _{i=1}^{Nc}{x}_i{b}_i}\end{array}\right.$$

(9)

For the SRK equation,

$$\begin{array}{l}u=1, w=0, \\ b_i=\frac{0.08664R{T}_{ci}}{P_{ci}}, \sqrt{a_i}={\left(\frac{0.42748}{P_{ci}}\right)}^{1/2}R{T}_{ci}\left[1+\left(1-\sqrt{T/T_{ci}}\right)f_{\omega }\right]\\ f_{\omega }=0.48+1.574{\omega }_i-0.176{\omega }_i^2\end{array}$$

(10)

For the PR equation,

$$\begin{array}{l}u=2, w=-1, \\ b_i=\frac{0.07780R{T}_{ci}}{P_{ci}}, \sqrt{a_i}={\left(\frac{0.45724}{P_{ci}}\right)}^{1/2}R{T}_{ci}\left[1+\left(1-\sqrt{T/T_{ci}}\right)f_{\omega }\right]\\ f_{\omega }=\left\{\begin{array}{l}0.37464+1.54226{\omega }_i-0.2699{\omega }_i^2 \left({\omega }_i\le 0.49\right)\\ 0.379642+1.48503{\omega }_i-0.164423{\omega }_i^2+0.016666{\omega }_i^3 \left({\omega }_i>0.49\right)\end{array}\right.\end{array}$$

(11)

2.3. Gas–Liquid State Judgment Model

The experimental findings demonstrate that miscible flooding is the most effective development approach for CO₂ flooding. Solving the component model of CO₂ is a complicated process, especially for miscible flooding. Due to the frequent changes in the gas, liquid, and miscible phases during gas injection, especially after each Newtonian step, the unknown quantity of the model is updated. At this time, it is necessary to judge the disappearance or reappearance of the gas phase or oil phase and calculate the composition and volume fraction of the new phase. This process is called phase equilibrium calculation, and both reservoir grids and well nodes require phase balance calculation. The CO₂ composition model’s phase equilibrium calculation involves two main steps: phase determination and P-T flash [23,24,25,26].

In this study, we utilized the tangential plane distance (TPD) function of the variation in Gibbs free energy to determine extreme values. The methodology involves determining whether a new phase can be segregated from a homogeneous hydrocarbon mixture, leading to a reduction in the overall system’s Gibbs free energy. The volume of the new phase is infinitesimal and does not affect the pressure and the composition of the original phase. Therefore, it is called the trail phase. The change in Gibbs free energy is defined as the TPD function.

$$tpd(w)={\displaystyle \sum _{i=1}^{Nc}{w}_i\left(\ln f_{i,g}\left(w\right)-d_i\right)}$$

(12)

where $w=\left(w_1,\cdots ,w_{Nh}\right)$, $d_i=\ln f_{i,o}\left(z\right)$, and $z=\left(z_1,\cdots ,z_{Nc}\right)$.

Phase determination involves determining whether the test phase is in the oil phase (when the test phase is gas) or gas phase (when the test phase is oil), thus establishing the phase state of the mixture. The complete phase stability test includes both a gas phase determination and a liquid phase determination. Take oil phase judgment as an example: if the minimum value of the TPD function is negative under fixed pressure, the original phase is unstable; otherwise, the original phase is stable.

To solve the extremum of the TPD function, the solution of the $tpd(w)$ function can be converted to the extremum of the following set of equations:

$$\ln W_T+\ln f_{ig}-d_i=0,\left(i=1,\cdots ,N_c\right)$$

(13)

where $W_T={\displaystyle \sum _{i=1}^{Nc}{W}_i}$ is referred to as the number of moles.

There are usually two methods to solve the above nonlinear equations: one is fixed-point iteration and the other is the Newton method. The fixed-point iteration constructed in this paper takes the equilibrium constant $K_i$ as the fixed point, which is also called successive substitution iteration (SSI). The iterative formula is as follows:

$$K_i^{n+1}=K_i^n\frac{d_i}{W_T{f}_{i,g}\left(w\right)},\left(i=1,\cdots ,N_c\right)$$

(14)

where $K_i=\frac{z_i}{W_i}$, $W_T={\displaystyle \sum _{i=1}^{Nc}{W}_i}$, $w_i=\frac{W_i}{W_T}$. The initial $K_i$ is estimated using Wilson’s formula:

$$K_i=\frac{{\left(p_{\mathrm{crit}}\right)}_i}{p}\exp \left[5.37(1+{\omega }_i)\left(1-\frac{{\left(T_{\mathrm{crit}}\right)}_i}{T}\right)\right]$$

(15)

SSI has global convergence, but its convergence speed is slow. The Principal Eigenvalue Acceleration (DEM) method is adopted to accelerate SSI. Furthermore, in order to solve the problem of slow convergence of SSI near the critical point, we define error $r_K={\displaystyle \sum _{i=1}^{Nc}\left|\frac{d_i}{W_T{f}_{i,g}\left(w\right)}-1\right|}$. It sometimes takes more than 1000 times for SSI to make $r_K<{10}^{-8}$ near the critical point. The Newton method has a second-order convergence rate but does not have global convergence, and it can only converge when the initial guess is good. Therefore, when searching in this study, the algorithm switches between SSI and Newton’s method, using SSI first when $r_K<{10}^{-2}$ is switched to the BFGS quasi-Newton method but switching back to SSI when the Newton method cannot reduce $r_K$. In order to make the Hessian matrix form of $tpd(w)$ concise, further substitution is made for variable $W_i$, so that the expression of the Hessian matrix for ${\alpha }_i=2\sqrt{W_i}$ is

$$H_{ij}=\frac{{\alpha }_i{\alpha }_j}{4W_T}\left[1+\frac{1}{f_{i,g}}\left(\frac{\partial f_{i,g}}{\partial w_j}-{\displaystyle \sum _{k=1}^{Nh}\frac{\partial f_{i,g}}{\partial w_k}{w}_k}\right)\right]$$

(16)

In the equation, $\frac{\partial f_{i,g}}{\partial w_j}$ is the partial derivative of fugacity with respect to the mole fraction. A useful feature of this formula is that when the fugacity is equivalent to the partial pressure $f_{i,g}=w_{i}p$, $H_{ij}={\delta }_{ij}$; that is, $H$ is the identity matrix.

The eigenvalues of the Hessian matrix can be used to determine whether a function has non-trivial extremum. Only when the minimum eigenvalue of $H_{ij}$ is positive does $tpd(w)$ have a non-trivial minimum.

Whether the reservoir fluid is in a single-phase, two-phase, or miscible-phase state can be determined by solving the extreme value of $tpd(w)$. If the minimum value of $tpd(w)$ is ≤0, this suggests a two-phase state. If the minimum value of $tpd(w)$ is >0, this indicates a single-phase state.

Then, whether the fluid in a grid is miscible can be ascertained based on the results of phase equilibrium calculations. If both the gas-phase stability test and the liquid-phase stability test are returned as ‘trivial’, $tpd(w)$ has no minimum value. Then, the mixture within this grid reaches the miscible phase.

The calculation process of obtaining the mole fraction of each phase when the total mole fraction $\left(z_1,\dots ,z_{Nc}\right)$ is known is called two-phase flash calculation. The goal is to find phase compositions that guarantee minimum and equal fugacity. If the phase state determination indicates that the hydrocarbon mixture will separate into two phases, or if the hydrocarbon mixture was two phases in the previous Newtonian step, it is necessary to calculate the mole fraction (L) of the oil phase and all $x_i$ and $y_i$ through flash calculation. In this study, the two-phase P-T flash evaporation calculates the component mole fractions of the gas and liquid phases assuming constant pressure and temperature. Its mathematical form is as follows: given $\left(z_1,\dots ,z_{Nc}\right)$ $p$ and $T$, then solve the following equation system:

$$\left\{\begin{array}{l}{f}_{i,o}\left(p,x_1,\dots ,x_{Nc}\right)=f_{i,g}\left(p,y_1,\dots ,y_{Nc}\right)\\ x_{i}L+y_i\left(1-L\right)=z_i\\ {\displaystyle \sum _{i=1}^{Nc}{x}_i}=1\end{array}\right.$$

(17)

According to the equilibrium constant $K_i$ obtained from the phase state determination, $K_i$ is taken as the initial value for the flash calculation, and the Rachford Rice equation is still solved using SSI and Newton methods. That is,

$${\displaystyle \sum _{i=1}^{Nc}\frac{z_i\left(K_i-1\right)}{L+\left(1-L\right)K_i}}=0$$

(18)

After obtaining $L$, $x_i$ and $y_i$ can be solved:

$$x_i=\frac{z_i}{L+\left(1-L\right)K_i}$$

(19)

$$y_i=\frac{K_i{z}_i}{L+\left(1-L\right)K_i}$$

(20)

2.4. IFT and Relative Permeability Model

The oil–gas interfacial tension is a measure of oil–gas miscibility. $\sigma$ is the oil–gas interfacial tension calculated using the Parachor model, which is closely related to the density of oil–gas components, and the formula is as follows:

$$\sigma ={\left[{\displaystyle \sum _{i=1}^{Nc}{\pi }_i\left({\rho }_o{x}_i-{\rho }_g{y}_i\right)}\right]}^4$$

(21)

In the two-phase region with oil and gas, as the mixture approaches the critical point, the density of oil and gas tends to be equal, and their interfacial tension tends to be zero. When the interfacial tension between oil and gas is zero, they no longer have a phase interface, thus becoming completely miscible; in the single-phase area, when the temperature passes through the critical temperature, the mixture changes continuously from oil phase to complete miscibility, and its relative permeability also changes continuously with the degree of miscibility. Therefore, this article constructs $k_X$ to represent this process, and the formula is as follows:

$$k_X=\lambda k_{X\_imm}+\left(1-\lambda \right)k_{X\_mis}$$

(22)

$X$ can be expressed as $row$, $rgw$, $rgo$, or $rog$, corresponding to the relative permeability function of oil–water, gas–water, gas–oil, or oil–gas coexistence, respectively. The subscript $\_imm$ indicates the relative permeability under the completely immiscible state, and $\_mis$ indicates the relative permeability under the completely miscible state; $\lambda$ is a quantity that characterizes the degree of grid miscibility, and $\lambda$ satisfies the following formula:

$$\lambda =\left\{\begin{array}{c}0,\hspace{1em}\mathrm{for} \sigma =0\\ {\left(\frac{\sigma }{{\sigma }_0}\right)}^{0.25},\hspace{1em}\mathrm{for} 0<\sigma <{\sigma }_0\\ \hspace{1em}1,\hspace{1em}\mathrm{for} \sigma >{\sigma }_0\end{array}\right.$$

(23)

When $\lambda \in \left[0,1\right]$ and when $\sigma >{\sigma }_0$, $\lambda =1$ represents a non-miscible state; when $\sigma =0$, $\lambda =0$ represents a miscible state; when $0<\sigma <{\sigma }_0$ and $0<\lambda <1$, the table is between mixed and non-miscible phases.

When oil and gas are in a miscible phase state, as the oil density and viscosity decrease, the residual oil saturation also gradually decreases, the oil–gas interface gradually weakens, and the permeability curves of oil and gas are elevated, thereby increasing the flow rate of the two-phase flow and achieving the effect of improving oil and gas recovery.

2.5. Linear Equation Solver

The linear equation solver is a tool used to solve complex systems like the seepage model of CCUS, which comprises multiple nonlinear partial differential equations. The pressure and saturation variable equations exhibit distinct properties and are closely interconnected.

Its matrix structure of discrete equation has the following distinctions compared with the traditional black oil model and others. It is strongly nonlinear, with highly non-uniform parameters such as pressure, saturation, and molar fraction. The matrix becomes large in scale and is severely singular. There is strong coupling between unknowns and between reservoir formation and wells.

A tailor-made multistage preconditioner is built in view of the above matrix structure characteristics to obtain the pressure, saturation, and molar fraction of every grid. The solver consists of AMG, ILU, GS iteration, and Krylov subspace iteration methods. A composite preconditioner is constructed by combining a non-expansive smoothing operator with a preconditioner (see Figure 1). This preconditioner can quickly and effectively improve the solving speed of the multiphase and multicomponent mathematical model by approximately 2–5 times [27,28,29,30].

3. Application and Validation

Based on the above-mentioned numerical simulation model, a multiphase and multicomponent numerical simulator is formed. Through a number of SPE standard examples and the test, comparison, and application of actual reservoirs in the field, it is shown that the simulator can be used well for the simulation of CO₂ flooding. For example, the high-pressure physical property matching and numerical simulation of CO₂-WAG injection have been conducted in a carbonate reservoir with low permeability and low viscosity in the Middle East.

The reservoir fluid has a CH₄ content of 36.6% and an intermediate hydrocarbon (C₂~C₆) content of 26.06%, belonging to a light oil reservoir. Based on indoor tests conducted on a single well, the saturation pressure of reservoir fluid under the original formation conditions was found to be 2682 psi, with a crude oil viscosity of 0.284 mPa s, a crude oil density of 0.6378 g/cm³, and a reservoir temperature of 265 ℉. For C20+, the molecular weight is 414 and the specific gravity is 0.9331.

3.1. Calculation of High-Pressure Physical Property Parameters

High-pressure physical property calculation is the basis of numerical simulation of CO₂ injection in reservoirs. Through the high-pressure physical property simulation calculation, the volume coefficient, density, viscosity, and expansion coefficient of underground hydrocarbon fluid and CO₂ fluid are obtained, and the critical parameters and binary interaction coefficients of each component are calculated as well as the minimum miscibility pressure of the fluid and other parameters.

When fitting and calculating the high-pressure physical properties of the reservoir, this component will be combined with other components that have similar properties into a single entity based on the principle of component aggregation. For the case of CO₂ injection study, CO₂ is taken as a single component. Therefore, the reservoir fluid is grouped into eight pseudo-components, namely, CO₂, C₁ + N₂, C₂ + H₂S, C₃–C₅, C₆ + C₇, C₈–C₁₀, C₁₁ to C₁₉, and C₂₀₊.

3.1.1. Constant Component Expansion Experiment Fitting

The phase state calculation model is used to fit the equal component expansion experiment after combination, and the changes in the fluid volume coefficient, density, gas-phase viscosity, and oil-phase viscosity of the reservoir are simulated (see Figure 2, Figure 3, Figure 4 and Figure 5). When the reservoir pressure drops from 8000 to 270 psi, the fitting absolute error of the volume coefficient ranges from 0 to 0.1573, and the fitting relative error ranges from 0 to 2.33%. When the reservoir pressure decreases from 8000 psi to above the saturation pressure, the absolute error in fitting crude oil density ranges from 0.000497 to 0.10049 g/cm³, and the relative error in fitting ranges from 0.0075% to 0.7039%. When the reservoir pressure drops from 2300 to 108 psi, the absolute error of the gas-phase viscosity fitting is 0.0002~0.0009 mPa·s, and the relative error of fitting is 1.14~6.31%. When the reservoir pressure drops from 2300 to 108 psi, the absolute error of the liquid viscosity fitting is in the range of 0.0005~0.02 mPa·s, and the relative error of the fitting is 0.19~4.02%. According to the above simulation results, the fitting precision of the constant component expansion experiment is high, which meets the requirements for the simulation calculation.

3.1.2. Swelling Test Fitting

It is imperative to investigate the impact of CO₂ injection on the reservoir fluid phase. The model mentioned above is utilized to accurately determine the saturation pressure, volume coefficients, and other parameters in swelling tests. Specifically, the bubble point pressure and expansion coefficient of the reservoir fluid vary as the mole fraction of the injected carbon dioxide changes (refer to Figure 6 and Figure 7). The results from the simulation indicate that the relative error in fitting the bubble point pressure across various carbon dioxide mole numbers falls within the range of 0.1% to 5.77%. Additionally, the relative fitting error for the expansion coefficient at different CO₂ mole numbers ranges from 0% to 2.37%. The simulation result, i.e., the saturation pressure and the volume coefficient of the swelling test, has high fitting precision and meets the requirements of component simulation calculation.

Based on the Lorentz–Bray–Clark viscosity calculation equation and the above fitting calculation of the parameters such as bubble point pressure, volume coefficient, density, and viscosity in the formation fluid and swelling tests of the reservoir, the critical parameters for each pseudo-component reflecting the actual fluid properties of the reservoir were obtained from this analysis (refer to

Table 1. Characteristic parameters of proposed reservoir components.

Composition of Pseudo-Components	Molar Mass/ (g/mol)	Critical Pressure/ MPa	Critical Temperature/ K	Critical Volume/ (m3/mol)	Acentric Factor/ 1
CO2	44.01	7.37	304.2	0.0939	0.225
C1 + N2	16.12	4.58	189.86	0.0988	0.0084
C2 + H2S	30.07	4.88	305.43	0.147	0.098
C3~C5	57.27	3.74	423.36	0.26	0.197
C6 + C7	91.65	3.06	551.73	0.364	0.332
C8~C10	119.16	2.45	615.61	0.464	0.468
C11~C19	192.23	1.85	642.22	0.804	0.623
C20+	414	1.52	793.43	2.1	1.048

and

Table 2. Binary interaction coefficient for the pseudo-components.

Composition of Pseudo-Components	CO2	C1 + N2	C2 + H2S	C3~C5	C6 + C7	C8~C10	C11~C19	C20+
CO2	0
C1 + N2	0.1	0
C2 + H2S	0.1	0.0013	0
C3~C5	0.1	0.0079	0.0027	0
C6 + C7	0.1	0.0144	0.0069	0.0009	0
C8~C10	0.11	0.0201	0.0111	0.0028	0.0005	0
C11~C19	0.11	0.0378	0.0252	0.0116	0.0059	0.0031	0
C20+	0.11	0.0743	0.0571	0.0362	0.0257	0.0192	0.0072	0

), serving as the foundation for subsequent numerical simulations on CO₂ miscible flooding.

3.1.3. Calculation of MMP

A one-dimensional numerical simulation model for slim-tube experiments was established using the parameters from the indoor tubule experiment (see

Table 3. Simulation parameters of thin tube experiment.

Parameter	Filling	Length (m)	Inner Diameter (cm)	Porosity (%)	Gas Permeability (10−3 μm2)
Values sand	Quartz	12.19	4.6	29	3000

). The grid size of the model is 100 × 1 × 1, the grid step in the X direction is 1.219 cm, the grid size in the Y and Z directions is 8 cm, the porosity is 29%, and the permeability is 3000 × 10⁻³ μm². The first section (1, 1, 1) of the model is an injection well, and the end (100, 1, 1) of the model is a production well. By simulating the recovery factor of the reservoir with the injection of 1.2 PV of CO₂ under various displacement pressures, the recovery factor curve can be plotted (refer to Figure 8). The pressure at which the curve reaches a turning point represents the minimum miscible pressure of the reservoir fluid.

The minimum miscibility pressure of crude oil mixed with carbon dioxide, as determined by the slim tube experiment, is 3730 psi. The calculated MMP of the PVT equation of state is 3747 psi, and the error is only 0.43% compared with the experimental result of the slim tube, which meets the requirements of numerical simulation accuracy. The simulation results of minimum miscibility pressure verify the reliability of the fluid model and the accuracy of the PVT phase equilibrium calculation.

3.2. Numerical Simulation of CO₂-WAG Displacement

A typical well group in the reservoir was chosen for a study of the numerical simulation mechanism of CO₂ injection and water flooding. The well group comprises three production wells and six injection wells, all of which are horizontal. The model has a total of 59,600 effective grids and eight components. In the model, the WAG cycle is six months; that is, water with 2000 STB/d is injected six months after gas injection with 10,000 Mscf/d. At the same time, Killough hysteresis model is adopted to simulate the hysteresis effect to the relative permeability curve.

The multiphase and multicomponent numerical simulator calculated 50 WAG cycles. Figure 9 shows the daily oil and gas production curve of one production well and the comparison with the results of commercial software A, which is typically used by reservoir engineers. Based on the comparison, the calculation results of this model show no significant deviation from those obtained using mainstream commercial software A.

History matching was utilized to determine the remaining oil distribution. Figure 10 is the prediction of the remaining oil distribution after 50 years of CO₂-WAG injection in this well group. Based on the remaining oil, a program was developed with injectors capable of delivering 4000 Mscf/day of CO₂ to predict the reservoir’s performance, focusing on the recovery factor and CO₂ storage rate.

Figure 11 shows that the recovery rate increases while the storage rate decreases when the reservoir pressure is higher than the MMP. In the early stage, miscibility occurs because of the high pressure, the injected CO₂ dissolves in oil, and the oil recovery is enhanced rapidly. Some of the CO₂ is produced alongside the miscible oil, and the storage of CO₂ decreases obviously. The storage rate of the CO₂ decreases drastically with the decline in the reservoir pressure when the pressure is below the MMP because of the CO₂ breakthrough. This program can be optimized to achieve the best development impact.

4. Conclusions

This paper discusses a set of numerical simulation techniques for simulating CO₂ flooding and storage, including multiphase and multicomponent mathematical models, CO₂ hydrocarbon EOS, a gas–liquid equilibrium discrimination model, an IFT and phase permeability model, and efficient linear-equation-solving algorithms. The numerical simulator formed using the above models and solver can fully describe the hydrocarbon fluid flow, phase changes, and miscible flooding mechanism and characteristics of CO₂-EOR. The calculation results are also consistent with commercial simulators.

The application of the models in a low-permeability and low-viscosity carbonate reservoir in the Middle East showed that the models and simulator meet the accuracy required for the numerical simulation of CO₂ injection for oil recovery and storage, and the results are highly reliable. The findings have guiding significance for studying the improvement of oil and gas recovery via CO₂ flooding and the quantification of CO₂ storage.