# A Simplified Capillary Bundle Model for CO2-Alternating-Water Injection Using an Equivalent Resistance Method

Authors: Wendong Wang; Fankun Meng; Yuliang Su; Lei Hou; Xueyu Geng; Yongmao Hao; Lei Li
Journal: Geofluids
 (2020)

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## Abstract

CO2-alternating-water injection is an effective way of enhancing recovery for low-permeability oil reservoirs. The injection process is one of the essential issues that are facing severe challenges because of the low permeability and poor pore space connectivity. Previous researchers mentioned that water injection ability could be decreased by around 20% after the CO2-flooding; hence, it is necessary to quantify the water injectivity variation during an alternated injection process. In this paper, a CO2 convection-diffusion model is established based on the seepage law of CO2 and dissipation effect. The relationship between the width of miscible flooding and injection time is defined. Besides, an equivalent resistance method is introduced for developing a capillary bundle model for featuring an unequal diameter for CO2 water vapor alternate flooding. CO2-oil and CO2-water interactions are analyzed using the new model. The effects of oil viscosity, pore throat ratio, CO2 slug size, and equivalent permeability of the capillary bundle on water injection are analyzed. The result indicates that water injection ability increases with the rise of CO2 slug size and equivalent permeability of the capillary bundle and decreases with the increase of viscosity and pore throat ratio.


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## Body

## 1. Introduction

The CO2-alternating-water injection method, used in the low-permeability reservoirs, has the advantages of both CO2 flooding and water flooding. It represents a promising method for improving the recovery of oil in the future. In 1957, Mobil Company has performed a pilot CO2-alternating-water injection test in North Pembina of Alberta, and then, several tests were conducted in Canada and the United States. In the United States, many tests have been performed because of the sufficient CO2 sources [1, 2], whereas in China, the first pilot field application was started by CNOOC in Qikou oilfield. The test showed that during the middle-late stage of the nonhomogeneous reservoir, the CO2-alternating-water injection could not only block the advantage flow channel but also contribute to the optimization of water injection [2]. Although its significant benefits, this method still has several problems. One serious problem is the decreasing of injection ability after CO2-alternating-water has been injected into the formation [3, 4]. Therefore, it is significanlty important to study the variation of injection ability with capillary bundle model during CO2-alternating-water flooding process.The CO2-alternating-water injection in the low-permeability reservoirs is a multiphase flow process. Early researchers have studied the seepage and flooding mechanism of multiphase fluid in the porous medium. Dong et al. presented the results of immiscible WAG injection in a water-wet micromodel and found that in a gas/oil/water system, stable oil layers were formed between the gas and water phases [5]. Due to gas bubbles were always surrounded by oil layers, direct gas/water displacements were not observed. Meanwhile, he found that the oil recovery declined greatly with WAG cycles. Wang and Dong developed an interacting triangular tube bundle model based on capillaries of equilateral triangle cross sections [6]. The relationship between the residual oil saturation and the complete capillary number (CA) was investigated, and the effects of the tube size, tube size distribution, and viscosity ratio on the magnitude of entrapment were also studied using this tube bundle model. Jovanović et al. presented both the stagnant film model and the moving film model to study the energy loss problem when the fluid flows in the capillary [7]. They considered the interfacial pressure drop and the frictional pressure drop with the tube wall, and the influence of the static and fluid boundary layer on the fluid seepage. As microscopic flow mechanics gradually became the focus of investigation, some researchers began to pay attention to the complex seepage mechanism of pore space in porous media. Some of those researchers have been focusing on the alternate injection of water and gas by pore structure by establishing a pore network model as a simulation platform. Piri developed a quasistatic pore network model by fully considering the seepage characteristics of oil, gas, and water three-phase flow in the pore throat [8]. The variation of saturation and relative permeability of each phase in the process of water seepage are compared with the existing experiment, and the data is well fitted. Yang et al. simulated the WAG displacement process in different wettable porous media from a microscopic point of view and analyzed the oil, gas, and water saturation and distribution during WAG displacement [9]. The changed law of position and displacement characteristics explains the microscopic mechanism of three-phase seepage in porous media. Helland pointed out that a bundle-of-triangular-tube model can reproduce the main characteristics of mixed-wet capillary pressure curves with hysteretic scanning loops [10].The displacement core test could be used to study the effect on the injection of CO2-alternating-water process. Jinan et al. designed an experimental unit for long core displacement by considering the necessity of similarities upon core length, the complex mechanism, and periodic physical-chemical reaction during multicontact miscible phases [11]. Zhao et al. analyzed the affecting factors of CO2 flooding, which include permeability, the relative density of crude oil, layer heterogeneity, interlayer heterogeneity, and layer fluid channeling by establishing the component numerical simulation model with Eclipse software [12, 13]. They concluded that it is appropriate to carry out CO2 flooding for homogeneous and lightweight reservoirs in low-permeability oil fields. Zahoor et al. mentioned that proper water-alternating-gas process design and implementation requires better knowledge of wettability and wettability variations in particular [14]. Wettability, in addition to influencing flow parameters, strongly affects other design parameters such as the volumes of water and gas required for injection, well spacing, etc. Some other researchers also used experimental methods to study the increasing difficulty in injection. Prieditis et al. and Kamath et al. conducted displacement tests on carbonate and limestone cores and analyzed the factors that cause a decrease of injectivity [15, 16]. Prieditis et al. also performed a field test based on laboratory work and used a simple model to predict the injectivity, but good results were not achieved [15]. Roper et al. used a compositional model to simulate and interpret an injectivity test conducted in the Mabee Field in the San Andres Formation, Martin County, Texas [17, 18]. They mentioned that the validation of compositional simulation as a means for interpreting field tests and developing improved predictions of reservoir injectivity performance and geostatistical techniques could be used successfully to characterize high heterogeneity in carbonate reservoirs for injectivity calculations. Yang et al. used core samples collected from tight formations to conduct a series of water-alternating-CO2 flooding experiments with different water-alternating-CO2 ratios and slug sizes [19]. They found that fluid injectivity is strongly dependent on slug size, water-alternating-CO2 ratio, and cycle time.Besides the experimental work, early researchers have done some theoretical modeling work. Li et al. derived the injectivity equation for water-alternating-gas injection based on similarity theory [20]. Zhou et al. analyzed the effects of injection pressure, injection speed, slug size, and gas-water slug ratio on oil displacement efficiency [21]. Yan et al. used a numerical simulation method to study the best injection parameters for CO2 gas water alternative drive [22]. Hu substituted the permeability of the capillary model into low-permeability equation and used superposition method to get total flow rate and threshold pressure gradient of the rock [23]. This model gives a theoretical description of low-permeability seepage characteristics. Several researchers have studied the injection capacity of different reservoirs and found that the injection volume, injection velocity, formation temperature, formation pressure segment size, and proportion have an impact on the final oil displacement efficiency and recovery factor [24–29]. Pizarro and Lake considered the influence of heterogeneity and autocorrelation of reservoir permeability distribution on the injectivity [30]. This model gives insight into why injectivity calculated from a core permeability average is frequently different from the injectivity manifested by the well in question. Yang et al. performed numerical simulations to history-match the experimental measurements and conduct sensitivity analysis on operational parameters and achieved good matching [19].Overall, researchers have done relatively thorough research on the decreasing injectivity happens in low-permeability reservoir during CO2 injection. However, due to the complexity of flow behavior of water-oil-gas, just a few researchers have explained the abnormal variance of injectivity during water-alternating-CO2 injection. In this paper, a new seepage diffusion equation is established by considering the CO2 diffusion and seepage in the reservoir. It could effectively reflect the variance of CO2 injectivity in water-alternating-CO2 injection process. The distribution of oil, CO2, and water in the capillary model with different diameters has been analyzed, and the influence of geology and fluid properties on injectivity has been provided.


## 2. CO2 Flooding Dynamics considering Mass Transfer

### 2.1. Mathematical Modeling

Assuming that the CO2 flooding process in the capillary model is a piston, the molecular diffusion, convective diffusion, and viscosity difference diffusion due to the difference of viscosity between CO2 and crude oil will occur at the interface between CO2 and crude oil. From a thermodynamic point of view, diffusion coefficients are functions of pressure, temperature, and composition, and driven by chemical potential difference; however, the proposed model is complex and hard to use [31]. Therefore, in this paper, the classical Fick’s law is utilized. According to the classical Fick’s law, molecular diffusion is mainly caused by the change of CO2 concentration in crude oil, and the diffusion coefficients are assumed to be constant. Convective diffusion is mainly affected by the complexity of the internal channels of the pores. The diffusion under the viscosity gradient is related to the difference of viscosity between CO2 and crude oil, and it is also affected by molecular diffusion and convection-diffusion [32]. The diffusion equation is derived from mass transfer equations by considering the balance of matter into and out of the capillary unit. To derive the diffusion equation of CO2 in the capillary, here, a πR2dx flowing unit of the capillary tube is studied (Figure 1).Figure 1
Flow unit of the capillary tube.According to the principle of material balance,theinfluentmass=effluentmass+amountofsubstanceaccumulatedintheunit, which is as follows:
(1)∂C∂t=−∂vD∂x−ω∂C∂x.According to the diffusion law,(2)vD=−D∂D∂x.Substituting Equation (2) into Equation (1), the material diffusion equation of the flow unit in the capillary is obtained.
(3)∂C∂t+ω∂C∂x=∂∂xD∂C∂x,where C is a unit concentration of CO2 after a mixed phase of crude oil and CO2, %; D is a comprehensive diffusion coefficient, cm2/s; ω is the flow velocity of the fluid in the capillary, cm/s. Here, the comprehensive diffusion coefficient D can be understood as a composite coefficient which not only considers the molecular diffusion and convection-diffusion of the single-phase fluid in the capillary but also considers the viscosity difference between the displacement and the displaced solution.For the method of determining the diffusion coefficient, based on the Stokes-Einstein relation, Kooijman developed a correlation and used UNIFAC parameters to correct for particle roundness and size. Hayduk and Minhas developed a set of correlations, specific to certain types of mixtures. Sigmund developed a correlation for high-pressure gas and liquid binary mixtures with the expression that related the reduced density-diffusion coefficient product to a third-degree polynomial function of reduced density. For nonideal binary mixtures, Riazi and Whitson developed a correlation to predict gas and liquid diffusion coefficient. Currently, the most widely used correlation to calculate diffusion coefficient is presented by Wilke and Chang and is the function of viscosity, which is used in this paper [33].The results of a displacement experiment in which a liquid is miscible with another liquid in the presence of differential pressure indicate that the best approximation of the overall diffusion coefficient is as follows:(4)D=DE1+Kμ∇μm,DE=Dm+Dμ,Dμ=Kωω,where Dm is the molecular diffusion coefficient, cm2/s; Du is the convection-diffusion coefficient of single-phase fluid, cm2/s; μm is the viscosity of two liquid mixtures, mPa·s; Kω and Kμ are the experimental coefficients which consider the experimental coefficients of single-phase fluid convection and different viscosity diffusion, with the unit of cm and cm/(MPa·s).From Equation (4),
(5)D=Dm+Du+Dm+DuKμ∇μm.Meanwhile,(6)Dμ=Dm+DuKμ∇μm.Dμ is the diffusion coefficient of viscosity difference between displacement fluid and displaced fluid. Equation (6) shows that the comprehensive diffusion coefficient is the sum of the molecular diffusion coefficient, the convection-diffusion coefficient, and the diffusion coefficient of the viscosity difference between the displacement fluid and the displaced fluid, which is a mixing coefficient.


### 2.2. Solution Strategy

In miscible region, we assume that there is only a single phase, which is the mixture of oil and gas phase. We introduced a new variable,τ=t,ξ=x−u/ϕ, into Equation (3). Then, Equation (3) become the standard diffusion equation.
(7)∂C∂τ=∂∂ξD∂C∂ξ,where ξ is called the self-modulation variable which changes with time and space.The boundary conditions are as follows:(8)C−λ,t=1,(9)Cλ,t=0,(10)C0,t=0.5,where ±λ is the miscible half-width, cm.Also, it is assumed that there is no CO2 inflow or outflow at the boundary of the miscible zone.
(11)∂C∂ξξ=±λ=0.The viscosity of the miscible zoneμm is also given in Figure 2 and is considered to be linear, so the viscosity of the miscible zone is as follows:
(12)∇μm=μ2−μ12λ.Figure 2
Schematic of CO2 flooding (1: CO2 region, 2: miscible region, and 3: oil region).Assumingβ=Kμμ2−μ1/2, then Kμ∇μm=β/λ. Equation (7) becomes the following:
(13)∂C∂τ=∂∂ξDE1+βλ∂C∂ξ.After solving Equation (13), the differential form solution can be obtained as follows:
(14)dλdτ=4DE1λ1+βλ.Thus, the corresponding “integral form solution” is given as follows:(15)λ22−βλ+β2lnλ+ββ=4DEτ.If the full length of the mixed-phase zone isΛ=2λ, Equation (15) can be transformed as follows:
(16)Λ24−βΛ+2β2lnΛ+2β2β=8DEτ.Equations (15) and (16) can be used to calculate the half-width λ or width Λ of the miscible region at any given timet, and λ=λt and Λ=Λt can be obtained by the iterative method that can be used to solve.


## 2.1. Mathematical Modeling

Assuming that the CO2 flooding process in the capillary model is a piston, the molecular diffusion, convective diffusion, and viscosity difference diffusion due to the difference of viscosity between CO2 and crude oil will occur at the interface between CO2 and crude oil. From a thermodynamic point of view, diffusion coefficients are functions of pressure, temperature, and composition, and driven by chemical potential difference; however, the proposed model is complex and hard to use [31]. Therefore, in this paper, the classical Fick’s law is utilized. According to the classical Fick’s law, molecular diffusion is mainly caused by the change of CO2 concentration in crude oil, and the diffusion coefficients are assumed to be constant. Convective diffusion is mainly affected by the complexity of the internal channels of the pores. The diffusion under the viscosity gradient is related to the difference of viscosity between CO2 and crude oil, and it is also affected by molecular diffusion and convection-diffusion [32]. The diffusion equation is derived from mass transfer equations by considering the balance of matter into and out of the capillary unit. To derive the diffusion equation of CO2 in the capillary, here, a πR2dx flowing unit of the capillary tube is studied (Figure 1).Figure 1
Flow unit of the capillary tube.According to the principle of material balance,theinfluentmass=effluentmass+amountofsubstanceaccumulatedintheunit, which is as follows:
(1)∂C∂t=−∂vD∂x−ω∂C∂x.According to the diffusion law,(2)vD=−D∂D∂x.Substituting Equation (2) into Equation (1), the material diffusion equation of the flow unit in the capillary is obtained.
(3)∂C∂t+ω∂C∂x=∂∂xD∂C∂x,where C is a unit concentration of CO2 after a mixed phase of crude oil and CO2, %; D is a comprehensive diffusion coefficient, cm2/s; ω is the flow velocity of the fluid in the capillary, cm/s. Here, the comprehensive diffusion coefficient D can be understood as a composite coefficient which not only considers the molecular diffusion and convection-diffusion of the single-phase fluid in the capillary but also considers the viscosity difference between the displacement and the displaced solution.For the method of determining the diffusion coefficient, based on the Stokes-Einstein relation, Kooijman developed a correlation and used UNIFAC parameters to correct for particle roundness and size. Hayduk and Minhas developed a set of correlations, specific to certain types of mixtures. Sigmund developed a correlation for high-pressure gas and liquid binary mixtures with the expression that related the reduced density-diffusion coefficient product to a third-degree polynomial function of reduced density. For nonideal binary mixtures, Riazi and Whitson developed a correlation to predict gas and liquid diffusion coefficient. Currently, the most widely used correlation to calculate diffusion coefficient is presented by Wilke and Chang and is the function of viscosity, which is used in this paper [33].The results of a displacement experiment in which a liquid is miscible with another liquid in the presence of differential pressure indicate that the best approximation of the overall diffusion coefficient is as follows:(4)D=DE1+Kμ∇μm,DE=Dm+Dμ,Dμ=Kωω,where Dm is the molecular diffusion coefficient, cm2/s; Du is the convection-diffusion coefficient of single-phase fluid, cm2/s; μm is the viscosity of two liquid mixtures, mPa·s; Kω and Kμ are the experimental coefficients which consider the experimental coefficients of single-phase fluid convection and different viscosity diffusion, with the unit of cm and cm/(MPa·s).From Equation (4),
(5)D=Dm+Du+Dm+DuKμ∇μm.Meanwhile,(6)Dμ=Dm+DuKμ∇μm.Dμ is the diffusion coefficient of viscosity difference between displacement fluid and displaced fluid. Equation (6) shows that the comprehensive diffusion coefficient is the sum of the molecular diffusion coefficient, the convection-diffusion coefficient, and the diffusion coefficient of the viscosity difference between the displacement fluid and the displaced fluid, which is a mixing coefficient.


## 2.2. Solution Strategy

In miscible region, we assume that there is only a single phase, which is the mixture of oil and gas phase. We introduced a new variable,τ=t,ξ=x−u/ϕ, into Equation (3). Then, Equation (3) become the standard diffusion equation.
(7)∂C∂τ=∂∂ξD∂C∂ξ,where ξ is called the self-modulation variable which changes with time and space.The boundary conditions are as follows:(8)C−λ,t=1,(9)Cλ,t=0,(10)C0,t=0.5,where ±λ is the miscible half-width, cm.Also, it is assumed that there is no CO2 inflow or outflow at the boundary of the miscible zone.
(11)∂C∂ξξ=±λ=0.The viscosity of the miscible zoneμm is also given in Figure 2 and is considered to be linear, so the viscosity of the miscible zone is as follows:
(12)∇μm=μ2−μ12λ.Figure 2
Schematic of CO2 flooding (1: CO2 region, 2: miscible region, and 3: oil region).Assumingβ=Kμμ2−μ1/2, then Kμ∇μm=β/λ. Equation (7) becomes the following:
(13)∂C∂τ=∂∂ξDE1+βλ∂C∂ξ.After solving Equation (13), the differential form solution can be obtained as follows:
(14)dλdτ=4DE1λ1+βλ.Thus, the corresponding “integral form solution” is given as follows:(15)λ22−βλ+β2lnλ+ββ=4DEτ.If the full length of the mixed-phase zone isΛ=2λ, Equation (15) can be transformed as follows:
(16)Λ24−βΛ+2β2lnΛ+2β2β=8DEτ.Equations (15) and (16) can be used to calculate the half-width λ or width Λ of the miscible region at any given timet, and λ=λt and Λ=Λt can be obtained by the iterative method that can be used to solve.


## 3. Unequal Diameter Capillary Beam Model for CO2-Alternating-Gas Flooding

### 3.1. Physical Model

It is assumed that the capillary beam model consists ofn capillaries with different capillary radii. Since in real oil reservoir, for any one flowing path, there are pores and throats; therefore, to reflect this pattern, for each capillary, the radius is not equal everywhere along the axis, and there is a throat. For the ith capillary, the pore throat structure is as follows in Figure 3.Figure 3
Structural schematic of pore and throat in the capillary tube.The capillary length isL, the throat position is Lc from the injection end of the capillary section, throat length is Lt, the capillary radius is Rp, and the radius at the throat is as follows:
(17)Rx=Rp+Rt2+Rp−Rt2cos2πxLt,where Rt is the throat radius, μm; Rx is the radius of the junction of the pore and the throat, μm; x is between 0~Lt, cm; the total volume of the throat is as follows:
(18)V=π∫0LtR2xdx.According to the definition of alternate driving of water and gas, the injection process can be divided into two stages, namely, CO2 injection and water injection. Because the throat is arbitrarily distributed for the same capillary, the throat position is various for different capillaries, and the fluid distribution in the capillary is complicated because of the difference in CO2 injection time.In the process of CO2 injection, six distribution patterns exist in the crude oil, mixed-phase zone, and CO2 in the capillary tube, and in the process of water flooding process, there are three fluid distribution patterns. Schematics and descriptions for those fluid distribution patterns are shown in Table 1.Table 1
Schematics for fluid distribution pattern during CO2 and water flooding.
ProcessNumberFluid distribution patternDescriptionCO2 floodingIThe front edge of the mixed-phase zone does not enter into the throat; the fluid in the throat is crude oil.IIThe front of the CO2 flooding is not entering the throat, but the right end of the miscible belt has entered the throat and occupied a part of the throat.IIIThe front of the CO2 flooding is not entering the throat, but the right end of the miscible belt has entered the throat and occupied a part of the throat.IVThe leading edge of the CO2 displacement has not entered the throat, the volume of the mixed phase is larger than the volume of the throat, and the right end has passed through the throat.VThe leading edge of the CO2 displacement enters into the throat, and the right end of the miscible zone passes through the throat.VIThe leading edge of CO2 displacement passes through the throat and the throat is filled with CO2.Water floodingVIIWater leading edge does not enter the throat.VIIIWater leading edge enters the throat.IXWater leading edge passes through the throat.


### 3.2. CO2 Flooding Seepage Mode and Mathematical Description

To facilitate the solution and analysis of the mathematical model, CO2/water is injected by constant pressure. The injection pressure of each capillary is equal, and the pressure difference between the injection end and the production end is ΔP. Introducing the seepage resistance coefficient to describe the magnitude of the seepage resistance and using the equivalent resistance model to solve the capillary beam injection flow rate are as follows [34]:


#### 3.2.1. Fluid Distribution Pattern I

For theith capillary, at the beginning of time j (j=1,2,3,⋯,Ntg), when xgij−1+xλij−1<Lc, the seepage resistance in the CO2 zone is as follows:
(19)Wgij=8μgxgij−1πRpi4,where xgij−1=0 is the length of CO2 region at the j−1 moment for the ith capillary, cm; for the beginning, xgi0=0. μg is the CO2 viscosity, mPa·s; Rpi is the capillary radius, μm.The seepage resistance of the miscible zone is as follows:(20)Wgij=8μ¯ogxλij−1πRpi4,where xλij−1 is the miscible length at the moment of j−1, cm; for the beginning, xgi0=0; μ¯og is the miscible fluid viscosity, mPa·s.The seepage resistance in the crude oil zone is as follows:(21)Wo,ij=8μ0L−xgij−1−xλij−1−LtiπRpi4+μ0∫0LtidxGx,where G is the conductivity of a fluid per unit length, μm4.
(22)Gx=π128Atπ+Rx4=π8R4x.In the case of a constant pressure difference, the injection flow rate is as follows:(23)qgij=ΔP+Pc,o−gWegj,where Weq,ij=Wg,ij+Wλ,Ij+Wo,ij, because there is no interfacial tension between CO2 and crude oil, Pc,o−g=0.During theΔt time, it is assumed that the fluid seepage velocity in the capillary remains constant.
(24)Wij=qijπRpi2.The convection-diffusion coefficient between CO2 and crude oil is as follows:
(25)DEij=Dm+KwWgij.According to the derivation in Section2.2, the relationship for the length of miscible band at time j is obtained, and the relevant parameters are substituted. Using the Newton iteration method, the length of miscible band at the end of time jcan be obtained. Since in Equation (1), C is defined as the unit concentration of CO2 in the mixture phase; it also can be seen as the volume ratio of CO2 in the unit bulk volume approximately, as shown in Figure 2. Therefore, assuming the volume of dissolved CO2 in the miscible zone occupies 1/2 of its total volume, the amount of CO2 dissolved in the mixed phase is as follows:
(26)Vmisij=12πRpi2xλij.The length of the CO2 zone formed at the beginning of the time j+1 is as follows:
(27)xgij=VgijπRpi2=xgij−1+qg,ijΔt−VmisijπRpi2.The total amount of CO2 injected into the bundle of tubes is as follows:
(28)Qgj=∑i=1nqg,ij.If the calculated sum of the CO2 zone xgij and the miscible zone xλij is less than Lci, iterative calculation of j+1 time can be performed according to the above steps.


#### 3.2.2. Fluid Distribution Pattern II

At the beginning ofjn moment, when the sum of the obtained CO2 zone and the miscible zone length is greater than Lci, but the volume of the miscible belt is smaller than the volume of the throat, that is xgijn−1+xλijn−1≥Lci,xgijn−1≤Lci,2Vmisijn−1<Vti. At the end of the jn−1 moment, the length of the miscible belt in the throat Lλikk=1,k=k+1 is as follows:
(29)xgijn−1+xλijn−1−LciRpi2=∫0LλikR2xdx.The Lobatto numerical integral formula can be used to solveLλik; xλijn−1 is the equivalent length of the miscible band, cm; the actual width of the miscible band is xλijn−1=Lci−xgijn−1+Lλik. Meanwhile, the actual length of the crude oil zone is xoijn−1=L−Lci−Lλik.The seepage resistance of the miscible zone includes two parts, one is the seepage resistance in the capillary at the left end of the throat and the other is the percolation resistance in the throat. The expression is as follows:(30)Wλ,ijn−1=8μ¯ogLci−xgijn−1πRpi4+μ¯og∫0Lλ,ikdxGx.The percolation resistance of crude oil and CO2 in the capillary is as follows:
(31)Wo,ijn=μ0∫Lλ,ikLtidxGx+8μ0L−Lci−LtiπRpi4,(32)Wg,ijn=8μoxgijn−1πRpi4.At the end ofjn moment, the injection amount of the ith capillary and the total injection amount of the capillary beam CO2 can be calculated by Equations (23) and (28). At this time, the leading edge of CO2 displacement is still outside the throat. Using Equations (27) and (26), the length of the CO2 zone, xgijn, and the length of the equivalent miscible zone CO2, xλijn, at the end of jn can be obtained (regardless of the length of the throat).


#### 3.2.3. Fluid Distribution Pattern III

At the beginning ofjn moment, when the sum of the lengths of the CO2 region and the miscible region is greater than Lci, that is, xgijn−1+xλijn−1≥Lci,2Vmistjn−1>Vn, it indicates that at the beginning of jn, the right end of the miscible band has begun to pass through the throat. The length of each zone and the seepage resistance are shown in Equations (30)–(32). The same iterative calculation procedure as in mode (2) is used to obtain the capillary tube injection flow rate.The length of the miscible zone formed at the right end of the throat isxrλim (m=1, m=m+1), and the expression is as follows:
(33)xrλim=xgijd+xλijd−Lci−∫0LtiR2xdxR2.The actual miscible band width isxλijd=Lci−xgijd+Lti+xrλim. xλijd is the length of the equivalent miscible region at the end of jd moment. The expression of seepage resistance in the miscible zone is as follows:
(34)Wλ,ijd+1=8μ¯ogLci−xgijd+xrλimπRpi4+μ¯og∫0LtdxGx.The seepage resistance of the CO2 zone and the crude oil zone in the capillary is as follows:
(35)Wgijd+1=8μgxgijdπRpi4,(36)Wo,ijd+1=8μoL−Lc−Lt−xrλimπRpi4.At this time, CO2 flooding interface is still outside the throat. Using Equations (27) and (26), the length of the CO2 zone at the end of jd+1 isxgijd+1, and the length of the equivalent miscible zone isxλ,ijd+1. At this time, it is necessary to judge the relative size of xgijd+1 and Lci. If xgijd+1≤Lci, the iterative calculation can be continued according to the steps in the distribution pattern II.


#### 3.2.4. Fluid Distribution Pattern IV

Let CO2 entering length is Lgia (a=1, a=a+1), and the expression is as follows:
(37)xgijm−LciRpi2=∫0LgiaR2xdx,where xgijm is the equivalent length of the CO2 zone, cm; actual length is xgijm=Lci+Lgia.It is assumed that after the CO2 flooding leading edge enters the throat, CO2 is no longer dissolved in the crude oil, and the equivalent length of the CO2 zone remains constant. At the end of jm, the seepage resistance in the CO2 zone is as follows:
(38)Wgijm+1=8μgLciπRpi4+μg∫0LgiadxGx.Since the volume of the miscible zone is smaller than the throat volume, the actual length of the miscible zone in the throat at the end ofjm is xλijm=LλIk−Lgia (k has been accumulated), where Lλik can be defined by the following:
(39)Rpi2xλijm=∫LgiaLλIkR2xdx,where xλijm is the end of jm, the equivalent length of the miscible band, cm.Since CO2 does not dissolve into the crude oil when the CO2 flooding leading edge is in the throat, the equivalent length of the miscible zone remains constant until the CO2 flooding leading edge passes through the throat. The seepage resistance of the miscible zone is as follows:
(40)Wλ,ijm+1=μ¯og∫LgiaLλikdxGx.The actual length of oil zone isxoijm=L−Lci−Lti+Lti−Lλik. The seepage resistance in the crude oil zone is as follows:
(41)Wo,ijm+1=8μoL−Lci−LtiπRpi4+μo∫LλikLtidxGx.At the end ofjm, the length of the formed CO2 zone is as follows:
(42)xgijm+1=xgijm+qg,ijm+1ΔtπRpi2.The amount of CO2 injected in the ith capillary and the total injection amount of the capillary bundle can be calculated by Equations (23) and (28).


#### 3.2.5. Fluid Distribution Pattern V

At this time, the length of the miscible band formed in the capillary at the right end of the throat isxrλib (b=1, b=b+1).
(43)xrλib=xλijm−∫LgiaLtiR2xdxRpi2.The actual width of the miscible band isxλijm=Lti−Lgia+xrλib. At the beginning of jm+1, the seepage resistance of the miscible zone is as follows:
(44)Wλ,ijm+1=8μ¯ogxrλibπRpi4+μ¯og∫LgiaLtidxGx.The actual length of oil zone isxoijm=L−Lci−Lti−xrλib, and the seepage resistance in the crude oil zone is as follows:
(45)Wo,ijm+1=8μoL−Lci−Lti−xrλibπRpi4.The seepage resistance in the CO2 zone is shown in Equation (38). The CO2 injection amount in the ith capillary and the total injection amount in the capillary bundle can be obtained by Equations (23) and (28). The equivalent length of the miscible zone is constant, and the length of the formed CO2 zone can be obtained by Equation (42).


#### 3.2.6. Fluid Distribution Pattern VI

In the capillary, at the right end of the throat, the length of CO2 region formed by xrgic (c=1, c=c+1) is as follows:
(46)qg,ijcΔt−∫LgiaLtiπR2xdxπRpi2=xrgic.At this moment, the CO2 displacement front is outside the throat and CO2 will continue to diffuse into the crude oil. At the beginning of jc+1, the actual length of the CO2 zone is xgijc=Lci+Lti+xrgic. The equivalent length of the miscible band is xλijc, cm. Thereafter, the equivalent length of the miscible zone is equal to the actual length, and the actual length of the crude oil zone is xoijc=L−xgijc−xλijc.At the beginning ofjc+1, the seepage resistance in the CO2 zone is as follows:
(47)Wg,ijc+1=8μgLci+xrgicπRpi4+μg∫0LtidxGx.The seepage resistances of the miscible zone and the crude oil zone are as follows:(48)Wλ,ijc+1=8μ¯ogxλijcπRpi4,(49)Wo,ijc+1=8μoL−xgijc−xλijcπRpi4.Knowing the magnitude of seepage resistance in each zone, the amount of CO2 injection in the ith capillary and capillary bundle can be calculated by using Equations (23) and (28). According to Equations (47)–(49), combined with the formula of the mixed-phase band in Section 2, the length of CO2 zone and the miscible zone can be obtained. Here, it should be noted that since the length of the miscible slug is constant when the leading edge of the CO2 displacement is assumed to be in the throat, the calculation of the miscible zone is performed after deducting the period of the CO2 displacement leading edge through the throat. In this way, according to the distribution pattern VI, the next iterative calculation is performed.


### 3.3. Water Flooding Seepage Mode and Mathematical Description

Suppose that the CO2 injection time is Tg, the time step has NtgNtg=Tg/Δt. At the beginning of Ntg+1, the water is injected with a constant pressure difference, ΔP. Assuming that CO2 is not dissolved in water, there is a gas-water interface. There is a capillary force, the CO2-water interfacial tension is σwg, and the contact angle is θw−g. At the beginning of water injection, when the leading edge of CO2 flooding is at the left end of the throat, during the subsequent water injection process, there may be six distribution patterns in the CO2 flooding process, but on the other hand, due to water injection, the following three fluid distribution patterns appear.


#### 3.3.1. Fluid Distribution Pattern VII

At this time, the seepage resistance of the water zone is as follows:(50)Ww,ij=8μwxwij−1πRpi4,where xwij−1 is the length of the water zone in the capillary at the beginning of Ntg+j, cm, xwi0=0.The pressure difference between the ends of the capillary bundle isΔP+Pci, which is kept constant, and the flow rate of injected water is as follows:
(51)qw,ij=ΔP+PciWeq,iNtg+j,Pci=2σw−gcosθw−gRpi,where Weq,iNtg+j=Wwij+Wg,iNtg+j+Wλ,iNtg+j+Wo,iNtg+j, Wg,iNtg+j,Wλ,iNig+j,Wo,iNtg+j is the magnitude of seepage resistance in the CO2 zone, the miscible zone, and the crude oil zone at the beginning of Ntg+j.The total amount of water injected is as follows:(52)QwNtg+j=∑i=1nqw,iNtg+j.During this process, the water seepage velocity is the same as CO2.
(53)Wwij=qw,ijπRpi2=WgiNtg+j.According to the principle of material balance, due to the constant total injection volume, CO2 will continue to diffuse into the crude oil during the water injection process, the length of the CO2 zone and the crude oil zone will decrease, the width of the miscible zone will increase, and the calculation formula of the miscible zone will be for the distribution mode. For the first, second, third, and sixth fluid distribution pattern, the formula for calculating the length of the CO2 zone should be corrected to the following:
(54)xgiNtg+j=xgiNtg+j−1−ΔVmisijπRpi2,where ΔVmistjis the amount of increase in the amount of CO2 dissolved in the miscible zone, cm3; xgiNtg+j and xgiNtg+j−1are the equivalent lengths of the CO2 zone at the time of Ntg+j and Ntg+j−1, cm.
(55)ΔVmisij=VmisiNtg+j−VmisiNtg+j−1=12πRpi2xλiNtg+j−xλiNtg+j−1,where xλiNtg+j and xλiNtg+j−1 are the equivalent lengths of the miscible band at the time of Ntg+j and Ntg+j−1. Since the diffusion of CO2 in water is not considered, the length of water zone in the capillary is as follows:
(56)xwij=xwij−1+qwijΔtπRpi2.


#### 3.3.2. Fluid Distribution Pattern VIII

Assuming at the beginning ofNtg+jw+1 moment (which is also at the end of Ntg+jw moment), the length of water zone is calculated as xwijw>Lci. It means that at the end of Ntg+jw, the injected water has entered the throat. The length needs to be corrected, and the actual water zone length isxwijw−1.
(57)xw,ijw=Lci+Lwid,(58)Rpi2xwijw−Lci=∫0LwidR2xdx,where xwijw is the equivalent length of the initial water zone at the time of Ntg+jw+1, cm. Lwid (d=1, d=d+1) is the extended length of the injected water in the throat.In the throat, the radius of the water-CO2 interface changes, so it will affect the capillary force. The capillary radius and capillary force at Lwid are as follows:
(59)RiLwid=RP+Rt2+Rp−Rt2cos2πLwidLt,(60)Pcid=2σw−gcosθ+γRix,(61)γ=tan−1dRxdx,(62)tanγ=dRxdx=RiLwid−RiLwid−1Lwid−Lwid−1,where RiLwi0=Rpi,Lwi0=0; γ is the correction value of the contact angle.At the beginning ofNtg+jw+1, the actual CO2 equivalent length is xgiNtg+jw, where the extension length at the right end of the throat is xrgie (e=1, e=e+1).
(63)xrgie=xgiNtg+jw−∫LwidLtiRx2dxRpi2.The seepage resistances in the water area and CO2 area are given in Equations (64) and (65).
(64)Ww,iNtg+jw+1=8μwLciπRpi4+μw∫0LwiddxGx,(65)Wg,iNtg+jw+1=8μgxrgieπRpi4+μg∫LwidLtidxGx.At this time, the length of the crude oil zone isL−Lci−Lti−xrgie−xλiNtg+jw, and the resistance of the crude oil zone can be obtained by substituting Equation (49), and the seepage resistance of the mixed phase can be obtained by using Equation (48). By substituting the formula for seepage resistance in each zone into Equations (51) and (52), the water injection amount of the ith capillary and capillary bundle can be obtained. According to Equations (51)-(55), the length of the water zone and the CO2 zone can be obtained, and the method of obtaining the miscible zone is the same as in previous cases.


#### 3.3.3. Fluid Distribution Pattern IX

At the beginning ofNtg+jg+1, if the calculated Lwid is greater than Lti, the water-CO2 interface has passed through the throat, and the length formed at the right end of throat isxwrie (e=1, e=e+1), xwrie=xwijg−Lci−∫0LtiRi2xdx/Rpi2..At the end ofNtg+jg, the actual length of the water area isxwijg=Lci+Lti+xwrie. xwijg is the equivalent length of water zone, the length of CO2 zone is xgiNig+jg, the length of miscible zone is xλiNig+jg, and the length of the crude oil zone is L−Lci−Lti−xλiNtg+jg−xgiNtg+jg−xwrie. Water-CO2 interfacial tension remains constant.At the beginning ofNtg+jg+1, the seepage resistance for each zone is as follows:
(66)Ww,iNtg+jg+1=8μwLci+xwrieπRI4+μw∫0LtidxGx,(67)Wg,iNtg+jg+1=8μgxgiNtg+jgπRi4,(68)Wλ,iNig+jg+1=8μ¯ogxλiNtg+jgπRi4,(69)Wo,iNtg+jg+1=8μoL−xwijg−xgiNtg+jg−xλiNtg+jgπRi4.Substituting Equations (66)-(69) into Equations (51)-(55), the length of the water zone and the CO2 zone can be obtained, and the parameters such as β and DE are substituted into Equation (40) to obtain the width of the miscible zone.


## 3.1. Physical Model

It is assumed that the capillary beam model consists ofn capillaries with different capillary radii. Since in real oil reservoir, for any one flowing path, there are pores and throats; therefore, to reflect this pattern, for each capillary, the radius is not equal everywhere along the axis, and there is a throat. For the ith capillary, the pore throat structure is as follows in Figure 3.Figure 3
Structural schematic of pore and throat in the capillary tube.The capillary length isL, the throat position is Lc from the injection end of the capillary section, throat length is Lt, the capillary radius is Rp, and the radius at the throat is as follows:
(17)Rx=Rp+Rt2+Rp−Rt2cos2πxLt,where Rt is the throat radius, μm; Rx is the radius of the junction of the pore and the throat, μm; x is between 0~Lt, cm; the total volume of the throat is as follows:
(18)V=π∫0LtR2xdx.According to the definition of alternate driving of water and gas, the injection process can be divided into two stages, namely, CO2 injection and water injection. Because the throat is arbitrarily distributed for the same capillary, the throat position is various for different capillaries, and the fluid distribution in the capillary is complicated because of the difference in CO2 injection time.In the process of CO2 injection, six distribution patterns exist in the crude oil, mixed-phase zone, and CO2 in the capillary tube, and in the process of water flooding process, there are three fluid distribution patterns. Schematics and descriptions for those fluid distribution patterns are shown in Table 1.Table 1
Schematics for fluid distribution pattern during CO2 and water flooding.
ProcessNumberFluid distribution patternDescriptionCO2 floodingIThe front edge of the mixed-phase zone does not enter into the throat; the fluid in the throat is crude oil.IIThe front of the CO2 flooding is not entering the throat, but the right end of the miscible belt has entered the throat and occupied a part of the throat.IIIThe front of the CO2 flooding is not entering the throat, but the right end of the miscible belt has entered the throat and occupied a part of the throat.IVThe leading edge of the CO2 displacement has not entered the throat, the volume of the mixed phase is larger than the volume of the throat, and the right end has passed through the throat.VThe leading edge of the CO2 displacement enters into the throat, and the right end of the miscible zone passes through the throat.VIThe leading edge of CO2 displacement passes through the throat and the throat is filled with CO2.Water floodingVIIWater leading edge does not enter the throat.VIIIWater leading edge enters the throat.IXWater leading edge passes through the throat.


## 3.2. CO2 Flooding Seepage Mode and Mathematical Description

To facilitate the solution and analysis of the mathematical model, CO2/water is injected by constant pressure. The injection pressure of each capillary is equal, and the pressure difference between the injection end and the production end is ΔP. Introducing the seepage resistance coefficient to describe the magnitude of the seepage resistance and using the equivalent resistance model to solve the capillary beam injection flow rate are as follows [34]:


### 3.2.1. Fluid Distribution Pattern I

For theith capillary, at the beginning of time j (j=1,2,3,⋯,Ntg), when xgij−1+xλij−1<Lc, the seepage resistance in the CO2 zone is as follows:
(19)Wgij=8μgxgij−1πRpi4,where xgij−1=0 is the length of CO2 region at the j−1 moment for the ith capillary, cm; for the beginning, xgi0=0. μg is the CO2 viscosity, mPa·s; Rpi is the capillary radius, μm.The seepage resistance of the miscible zone is as follows:(20)Wgij=8μ¯ogxλij−1πRpi4,where xλij−1 is the miscible length at the moment of j−1, cm; for the beginning, xgi0=0; μ¯og is the miscible fluid viscosity, mPa·s.The seepage resistance in the crude oil zone is as follows:(21)Wo,ij=8μ0L−xgij−1−xλij−1−LtiπRpi4+μ0∫0LtidxGx,where G is the conductivity of a fluid per unit length, μm4.
(22)Gx=π128Atπ+Rx4=π8R4x.In the case of a constant pressure difference, the injection flow rate is as follows:(23)qgij=ΔP+Pc,o−gWegj,where Weq,ij=Wg,ij+Wλ,Ij+Wo,ij, because there is no interfacial tension between CO2 and crude oil, Pc,o−g=0.During theΔt time, it is assumed that the fluid seepage velocity in the capillary remains constant.
(24)Wij=qijπRpi2.The convection-diffusion coefficient between CO2 and crude oil is as follows:
(25)DEij=Dm+KwWgij.According to the derivation in Section2.2, the relationship for the length of miscible band at time j is obtained, and the relevant parameters are substituted. Using the Newton iteration method, the length of miscible band at the end of time jcan be obtained. Since in Equation (1), C is defined as the unit concentration of CO2 in the mixture phase; it also can be seen as the volume ratio of CO2 in the unit bulk volume approximately, as shown in Figure 2. Therefore, assuming the volume of dissolved CO2 in the miscible zone occupies 1/2 of its total volume, the amount of CO2 dissolved in the mixed phase is as follows:
(26)Vmisij=12πRpi2xλij.The length of the CO2 zone formed at the beginning of the time j+1 is as follows:
(27)xgij=VgijπRpi2=xgij−1+qg,ijΔt−VmisijπRpi2.The total amount of CO2 injected into the bundle of tubes is as follows:
(28)Qgj=∑i=1nqg,ij.If the calculated sum of the CO2 zone xgij and the miscible zone xλij is less than Lci, iterative calculation of j+1 time can be performed according to the above steps.


### 3.2.2. Fluid Distribution Pattern II

At the beginning ofjn moment, when the sum of the obtained CO2 zone and the miscible zone length is greater than Lci, but the volume of the miscible belt is smaller than the volume of the throat, that is xgijn−1+xλijn−1≥Lci,xgijn−1≤Lci,2Vmisijn−1<Vti. At the end of the jn−1 moment, the length of the miscible belt in the throat Lλikk=1,k=k+1 is as follows:
(29)xgijn−1+xλijn−1−LciRpi2=∫0LλikR2xdx.The Lobatto numerical integral formula can be used to solveLλik; xλijn−1 is the equivalent length of the miscible band, cm; the actual width of the miscible band is xλijn−1=Lci−xgijn−1+Lλik. Meanwhile, the actual length of the crude oil zone is xoijn−1=L−Lci−Lλik.The seepage resistance of the miscible zone includes two parts, one is the seepage resistance in the capillary at the left end of the throat and the other is the percolation resistance in the throat. The expression is as follows:(30)Wλ,ijn−1=8μ¯ogLci−xgijn−1πRpi4+μ¯og∫0Lλ,ikdxGx.The percolation resistance of crude oil and CO2 in the capillary is as follows:
(31)Wo,ijn=μ0∫Lλ,ikLtidxGx+8μ0L−Lci−LtiπRpi4,(32)Wg,ijn=8μoxgijn−1πRpi4.At the end ofjn moment, the injection amount of the ith capillary and the total injection amount of the capillary beam CO2 can be calculated by Equations (23) and (28). At this time, the leading edge of CO2 displacement is still outside the throat. Using Equations (27) and (26), the length of the CO2 zone, xgijn, and the length of the equivalent miscible zone CO2, xλijn, at the end of jn can be obtained (regardless of the length of the throat).


### 3.2.3. Fluid Distribution Pattern III

At the beginning ofjn moment, when the sum of the lengths of the CO2 region and the miscible region is greater than Lci, that is, xgijn−1+xλijn−1≥Lci,2Vmistjn−1>Vn, it indicates that at the beginning of jn, the right end of the miscible band has begun to pass through the throat. The length of each zone and the seepage resistance are shown in Equations (30)–(32). The same iterative calculation procedure as in mode (2) is used to obtain the capillary tube injection flow rate.The length of the miscible zone formed at the right end of the throat isxrλim (m=1, m=m+1), and the expression is as follows:
(33)xrλim=xgijd+xλijd−Lci−∫0LtiR2xdxR2.The actual miscible band width isxλijd=Lci−xgijd+Lti+xrλim. xλijd is the length of the equivalent miscible region at the end of jd moment. The expression of seepage resistance in the miscible zone is as follows:
(34)Wλ,ijd+1=8μ¯ogLci−xgijd+xrλimπRpi4+μ¯og∫0LtdxGx.The seepage resistance of the CO2 zone and the crude oil zone in the capillary is as follows:
(35)Wgijd+1=8μgxgijdπRpi4,(36)Wo,ijd+1=8μoL−Lc−Lt−xrλimπRpi4.At this time, CO2 flooding interface is still outside the throat. Using Equations (27) and (26), the length of the CO2 zone at the end of jd+1 isxgijd+1, and the length of the equivalent miscible zone isxλ,ijd+1. At this time, it is necessary to judge the relative size of xgijd+1 and Lci. If xgijd+1≤Lci, the iterative calculation can be continued according to the steps in the distribution pattern II.


### 3.2.4. Fluid Distribution Pattern IV

Let CO2 entering length is Lgia (a=1, a=a+1), and the expression is as follows:
(37)xgijm−LciRpi2=∫0LgiaR2xdx,where xgijm is the equivalent length of the CO2 zone, cm; actual length is xgijm=Lci+Lgia.It is assumed that after the CO2 flooding leading edge enters the throat, CO2 is no longer dissolved in the crude oil, and the equivalent length of the CO2 zone remains constant. At the end of jm, the seepage resistance in the CO2 zone is as follows:
(38)Wgijm+1=8μgLciπRpi4+μg∫0LgiadxGx.Since the volume of the miscible zone is smaller than the throat volume, the actual length of the miscible zone in the throat at the end ofjm is xλijm=LλIk−Lgia (k has been accumulated), where Lλik can be defined by the following:
(39)Rpi2xλijm=∫LgiaLλIkR2xdx,where xλijm is the end of jm, the equivalent length of the miscible band, cm.Since CO2 does not dissolve into the crude oil when the CO2 flooding leading edge is in the throat, the equivalent length of the miscible zone remains constant until the CO2 flooding leading edge passes through the throat. The seepage resistance of the miscible zone is as follows:
(40)Wλ,ijm+1=μ¯og∫LgiaLλikdxGx.The actual length of oil zone isxoijm=L−Lci−Lti+Lti−Lλik. The seepage resistance in the crude oil zone is as follows:
(41)Wo,ijm+1=8μoL−Lci−LtiπRpi4+μo∫LλikLtidxGx.At the end ofjm, the length of the formed CO2 zone is as follows:
(42)xgijm+1=xgijm+qg,ijm+1ΔtπRpi2.The amount of CO2 injected in the ith capillary and the total injection amount of the capillary bundle can be calculated by Equations (23) and (28).


### 3.2.5. Fluid Distribution Pattern V

At this time, the length of the miscible band formed in the capillary at the right end of the throat isxrλib (b=1, b=b+1).
(43)xrλib=xλijm−∫LgiaLtiR2xdxRpi2.The actual width of the miscible band isxλijm=Lti−Lgia+xrλib. At the beginning of jm+1, the seepage resistance of the miscible zone is as follows:
(44)Wλ,ijm+1=8μ¯ogxrλibπRpi4+μ¯og∫LgiaLtidxGx.The actual length of oil zone isxoijm=L−Lci−Lti−xrλib, and the seepage resistance in the crude oil zone is as follows:
(45)Wo,ijm+1=8μoL−Lci−Lti−xrλibπRpi4.The seepage resistance in the CO2 zone is shown in Equation (38). The CO2 injection amount in the ith capillary and the total injection amount in the capillary bundle can be obtained by Equations (23) and (28). The equivalent length of the miscible zone is constant, and the length of the formed CO2 zone can be obtained by Equation (42).


### 3.2.6. Fluid Distribution Pattern VI

In the capillary, at the right end of the throat, the length of CO2 region formed by xrgic (c=1, c=c+1) is as follows:
(46)qg,ijcΔt−∫LgiaLtiπR2xdxπRpi2=xrgic.At this moment, the CO2 displacement front is outside the throat and CO2 will continue to diffuse into the crude oil. At the beginning of jc+1, the actual length of the CO2 zone is xgijc=Lci+Lti+xrgic. The equivalent length of the miscible band is xλijc, cm. Thereafter, the equivalent length of the miscible zone is equal to the actual length, and the actual length of the crude oil zone is xoijc=L−xgijc−xλijc.At the beginning ofjc+1, the seepage resistance in the CO2 zone is as follows:
(47)Wg,ijc+1=8μgLci+xrgicπRpi4+μg∫0LtidxGx.The seepage resistances of the miscible zone and the crude oil zone are as follows:(48)Wλ,ijc+1=8μ¯ogxλijcπRpi4,(49)Wo,ijc+1=8μoL−xgijc−xλijcπRpi4.Knowing the magnitude of seepage resistance in each zone, the amount of CO2 injection in the ith capillary and capillary bundle can be calculated by using Equations (23) and (28). According to Equations (47)–(49), combined with the formula of the mixed-phase band in Section 2, the length of CO2 zone and the miscible zone can be obtained. Here, it should be noted that since the length of the miscible slug is constant when the leading edge of the CO2 displacement is assumed to be in the throat, the calculation of the miscible zone is performed after deducting the period of the CO2 displacement leading edge through the throat. In this way, according to the distribution pattern VI, the next iterative calculation is performed.


## 3.2.1. Fluid Distribution Pattern I

For theith capillary, at the beginning of time j (j=1,2,3,⋯,Ntg), when xgij−1+xλij−1<Lc, the seepage resistance in the CO2 zone is as follows:
(19)Wgij=8μgxgij−1πRpi4,where xgij−1=0 is the length of CO2 region at the j−1 moment for the ith capillary, cm; for the beginning, xgi0=0. μg is the CO2 viscosity, mPa·s; Rpi is the capillary radius, μm.The seepage resistance of the miscible zone is as follows:(20)Wgij=8μ¯ogxλij−1πRpi4,where xλij−1 is the miscible length at the moment of j−1, cm; for the beginning, xgi0=0; μ¯og is the miscible fluid viscosity, mPa·s.The seepage resistance in the crude oil zone is as follows:(21)Wo,ij=8μ0L−xgij−1−xλij−1−LtiπRpi4+μ0∫0LtidxGx,where G is the conductivity of a fluid per unit length, μm4.
(22)Gx=π128Atπ+Rx4=π8R4x.In the case of a constant pressure difference, the injection flow rate is as follows:(23)qgij=ΔP+Pc,o−gWegj,where Weq,ij=Wg,ij+Wλ,Ij+Wo,ij, because there is no interfacial tension between CO2 and crude oil, Pc,o−g=0.During theΔt time, it is assumed that the fluid seepage velocity in the capillary remains constant.
(24)Wij=qijπRpi2.The convection-diffusion coefficient between CO2 and crude oil is as follows:
(25)DEij=Dm+KwWgij.According to the derivation in Section2.2, the relationship for the length of miscible band at time j is obtained, and the relevant parameters are substituted. Using the Newton iteration method, the length of miscible band at the end of time jcan be obtained. Since in Equation (1), C is defined as the unit concentration of CO2 in the mixture phase; it also can be seen as the volume ratio of CO2 in the unit bulk volume approximately, as shown in Figure 2. Therefore, assuming the volume of dissolved CO2 in the miscible zone occupies 1/2 of its total volume, the amount of CO2 dissolved in the mixed phase is as follows:
(26)Vmisij=12πRpi2xλij.The length of the CO2 zone formed at the beginning of the time j+1 is as follows:
(27)xgij=VgijπRpi2=xgij−1+qg,ijΔt−VmisijπRpi2.The total amount of CO2 injected into the bundle of tubes is as follows:
(28)Qgj=∑i=1nqg,ij.If the calculated sum of the CO2 zone xgij and the miscible zone xλij is less than Lci, iterative calculation of j+1 time can be performed according to the above steps.


## 3.2.2. Fluid Distribution Pattern II

At the beginning ofjn moment, when the sum of the obtained CO2 zone and the miscible zone length is greater than Lci, but the volume of the miscible belt is smaller than the volume of the throat, that is xgijn−1+xλijn−1≥Lci,xgijn−1≤Lci,2Vmisijn−1<Vti. At the end of the jn−1 moment, the length of the miscible belt in the throat Lλikk=1,k=k+1 is as follows:
(29)xgijn−1+xλijn−1−LciRpi2=∫0LλikR2xdx.The Lobatto numerical integral formula can be used to solveLλik; xλijn−1 is the equivalent length of the miscible band, cm; the actual width of the miscible band is xλijn−1=Lci−xgijn−1+Lλik. Meanwhile, the actual length of the crude oil zone is xoijn−1=L−Lci−Lλik.The seepage resistance of the miscible zone includes two parts, one is the seepage resistance in the capillary at the left end of the throat and the other is the percolation resistance in the throat. The expression is as follows:(30)Wλ,ijn−1=8μ¯ogLci−xgijn−1πRpi4+μ¯og∫0Lλ,ikdxGx.The percolation resistance of crude oil and CO2 in the capillary is as follows:
(31)Wo,ijn=μ0∫Lλ,ikLtidxGx+8μ0L−Lci−LtiπRpi4,(32)Wg,ijn=8μoxgijn−1πRpi4.At the end ofjn moment, the injection amount of the ith capillary and the total injection amount of the capillary beam CO2 can be calculated by Equations (23) and (28). At this time, the leading edge of CO2 displacement is still outside the throat. Using Equations (27) and (26), the length of the CO2 zone, xgijn, and the length of the equivalent miscible zone CO2, xλijn, at the end of jn can be obtained (regardless of the length of the throat).


## 3.2.3. Fluid Distribution Pattern III

At the beginning ofjn moment, when the sum of the lengths of the CO2 region and the miscible region is greater than Lci, that is, xgijn−1+xλijn−1≥Lci,2Vmistjn−1>Vn, it indicates that at the beginning of jn, the right end of the miscible band has begun to pass through the throat. The length of each zone and the seepage resistance are shown in Equations (30)–(32). The same iterative calculation procedure as in mode (2) is used to obtain the capillary tube injection flow rate.The length of the miscible zone formed at the right end of the throat isxrλim (m=1, m=m+1), and the expression is as follows:
(33)xrλim=xgijd+xλijd−Lci−∫0LtiR2xdxR2.The actual miscible band width isxλijd=Lci−xgijd+Lti+xrλim. xλijd is the length of the equivalent miscible region at the end of jd moment. The expression of seepage resistance in the miscible zone is as follows:
(34)Wλ,ijd+1=8μ¯ogLci−xgijd+xrλimπRpi4+μ¯og∫0LtdxGx.The seepage resistance of the CO2 zone and the crude oil zone in the capillary is as follows:
(35)Wgijd+1=8μgxgijdπRpi4,(36)Wo,ijd+1=8μoL−Lc−Lt−xrλimπRpi4.At this time, CO2 flooding interface is still outside the throat. Using Equations (27) and (26), the length of the CO2 zone at the end of jd+1 isxgijd+1, and the length of the equivalent miscible zone isxλ,ijd+1. At this time, it is necessary to judge the relative size of xgijd+1 and Lci. If xgijd+1≤Lci, the iterative calculation can be continued according to the steps in the distribution pattern II.


## 3.2.4. Fluid Distribution Pattern IV

Let CO2 entering length is Lgia (a=1, a=a+1), and the expression is as follows:
(37)xgijm−LciRpi2=∫0LgiaR2xdx,where xgijm is the equivalent length of the CO2 zone, cm; actual length is xgijm=Lci+Lgia.It is assumed that after the CO2 flooding leading edge enters the throat, CO2 is no longer dissolved in the crude oil, and the equivalent length of the CO2 zone remains constant. At the end of jm, the seepage resistance in the CO2 zone is as follows:
(38)Wgijm+1=8μgLciπRpi4+μg∫0LgiadxGx.Since the volume of the miscible zone is smaller than the throat volume, the actual length of the miscible zone in the throat at the end ofjm is xλijm=LλIk−Lgia (k has been accumulated), where Lλik can be defined by the following:
(39)Rpi2xλijm=∫LgiaLλIkR2xdx,where xλijm is the end of jm, the equivalent length of the miscible band, cm.Since CO2 does not dissolve into the crude oil when the CO2 flooding leading edge is in the throat, the equivalent length of the miscible zone remains constant until the CO2 flooding leading edge passes through the throat. The seepage resistance of the miscible zone is as follows:
(40)Wλ,ijm+1=μ¯og∫LgiaLλikdxGx.The actual length of oil zone isxoijm=L−Lci−Lti+Lti−Lλik. The seepage resistance in the crude oil zone is as follows:
(41)Wo,ijm+1=8μoL−Lci−LtiπRpi4+μo∫LλikLtidxGx.At the end ofjm, the length of the formed CO2 zone is as follows:
(42)xgijm+1=xgijm+qg,ijm+1ΔtπRpi2.The amount of CO2 injected in the ith capillary and the total injection amount of the capillary bundle can be calculated by Equations (23) and (28).


## 3.2.5. Fluid Distribution Pattern V

At this time, the length of the miscible band formed in the capillary at the right end of the throat isxrλib (b=1, b=b+1).
(43)xrλib=xλijm−∫LgiaLtiR2xdxRpi2.The actual width of the miscible band isxλijm=Lti−Lgia+xrλib. At the beginning of jm+1, the seepage resistance of the miscible zone is as follows:
(44)Wλ,ijm+1=8μ¯ogxrλibπRpi4+μ¯og∫LgiaLtidxGx.The actual length of oil zone isxoijm=L−Lci−Lti−xrλib, and the seepage resistance in the crude oil zone is as follows:
(45)Wo,ijm+1=8μoL−Lci−Lti−xrλibπRpi4.The seepage resistance in the CO2 zone is shown in Equation (38). The CO2 injection amount in the ith capillary and the total injection amount in the capillary bundle can be obtained by Equations (23) and (28). The equivalent length of the miscible zone is constant, and the length of the formed CO2 zone can be obtained by Equation (42).


## 3.2.6. Fluid Distribution Pattern VI

In the capillary, at the right end of the throat, the length of CO2 region formed by xrgic (c=1, c=c+1) is as follows:
(46)qg,ijcΔt−∫LgiaLtiπR2xdxπRpi2=xrgic.At this moment, the CO2 displacement front is outside the throat and CO2 will continue to diffuse into the crude oil. At the beginning of jc+1, the actual length of the CO2 zone is xgijc=Lci+Lti+xrgic. The equivalent length of the miscible band is xλijc, cm. Thereafter, the equivalent length of the miscible zone is equal to the actual length, and the actual length of the crude oil zone is xoijc=L−xgijc−xλijc.At the beginning ofjc+1, the seepage resistance in the CO2 zone is as follows:
(47)Wg,ijc+1=8μgLci+xrgicπRpi4+μg∫0LtidxGx.The seepage resistances of the miscible zone and the crude oil zone are as follows:(48)Wλ,ijc+1=8μ¯ogxλijcπRpi4,(49)Wo,ijc+1=8μoL−xgijc−xλijcπRpi4.Knowing the magnitude of seepage resistance in each zone, the amount of CO2 injection in the ith capillary and capillary bundle can be calculated by using Equations (23) and (28). According to Equations (47)–(49), combined with the formula of the mixed-phase band in Section 2, the length of CO2 zone and the miscible zone can be obtained. Here, it should be noted that since the length of the miscible slug is constant when the leading edge of the CO2 displacement is assumed to be in the throat, the calculation of the miscible zone is performed after deducting the period of the CO2 displacement leading edge through the throat. In this way, according to the distribution pattern VI, the next iterative calculation is performed.


## 3.3. Water Flooding Seepage Mode and Mathematical Description

Suppose that the CO2 injection time is Tg, the time step has NtgNtg=Tg/Δt. At the beginning of Ntg+1, the water is injected with a constant pressure difference, ΔP. Assuming that CO2 is not dissolved in water, there is a gas-water interface. There is a capillary force, the CO2-water interfacial tension is σwg, and the contact angle is θw−g. At the beginning of water injection, when the leading edge of CO2 flooding is at the left end of the throat, during the subsequent water injection process, there may be six distribution patterns in the CO2 flooding process, but on the other hand, due to water injection, the following three fluid distribution patterns appear.


### 3.3.1. Fluid Distribution Pattern VII

At this time, the seepage resistance of the water zone is as follows:(50)Ww,ij=8μwxwij−1πRpi4,where xwij−1 is the length of the water zone in the capillary at the beginning of Ntg+j, cm, xwi0=0.The pressure difference between the ends of the capillary bundle isΔP+Pci, which is kept constant, and the flow rate of injected water is as follows:
(51)qw,ij=ΔP+PciWeq,iNtg+j,Pci=2σw−gcosθw−gRpi,where Weq,iNtg+j=Wwij+Wg,iNtg+j+Wλ,iNtg+j+Wo,iNtg+j, Wg,iNtg+j,Wλ,iNig+j,Wo,iNtg+j is the magnitude of seepage resistance in the CO2 zone, the miscible zone, and the crude oil zone at the beginning of Ntg+j.The total amount of water injected is as follows:(52)QwNtg+j=∑i=1nqw,iNtg+j.During this process, the water seepage velocity is the same as CO2.
(53)Wwij=qw,ijπRpi2=WgiNtg+j.According to the principle of material balance, due to the constant total injection volume, CO2 will continue to diffuse into the crude oil during the water injection process, the length of the CO2 zone and the crude oil zone will decrease, the width of the miscible zone will increase, and the calculation formula of the miscible zone will be for the distribution mode. For the first, second, third, and sixth fluid distribution pattern, the formula for calculating the length of the CO2 zone should be corrected to the following:
(54)xgiNtg+j=xgiNtg+j−1−ΔVmisijπRpi2,where ΔVmistjis the amount of increase in the amount of CO2 dissolved in the miscible zone, cm3; xgiNtg+j and xgiNtg+j−1are the equivalent lengths of the CO2 zone at the time of Ntg+j and Ntg+j−1, cm.
(55)ΔVmisij=VmisiNtg+j−VmisiNtg+j−1=12πRpi2xλiNtg+j−xλiNtg+j−1,where xλiNtg+j and xλiNtg+j−1 are the equivalent lengths of the miscible band at the time of Ntg+j and Ntg+j−1. Since the diffusion of CO2 in water is not considered, the length of water zone in the capillary is as follows:
(56)xwij=xwij−1+qwijΔtπRpi2.


### 3.3.2. Fluid Distribution Pattern VIII

Assuming at the beginning ofNtg+jw+1 moment (which is also at the end of Ntg+jw moment), the length of water zone is calculated as xwijw>Lci. It means that at the end of Ntg+jw, the injected water has entered the throat. The length needs to be corrected, and the actual water zone length isxwijw−1.
(57)xw,ijw=Lci+Lwid,(58)Rpi2xwijw−Lci=∫0LwidR2xdx,where xwijw is the equivalent length of the initial water zone at the time of Ntg+jw+1, cm. Lwid (d=1, d=d+1) is the extended length of the injected water in the throat.In the throat, the radius of the water-CO2 interface changes, so it will affect the capillary force. The capillary radius and capillary force at Lwid are as follows:
(59)RiLwid=RP+Rt2+Rp−Rt2cos2πLwidLt,(60)Pcid=2σw−gcosθ+γRix,(61)γ=tan−1dRxdx,(62)tanγ=dRxdx=RiLwid−RiLwid−1Lwid−Lwid−1,where RiLwi0=Rpi,Lwi0=0; γ is the correction value of the contact angle.At the beginning ofNtg+jw+1, the actual CO2 equivalent length is xgiNtg+jw, where the extension length at the right end of the throat is xrgie (e=1, e=e+1).
(63)xrgie=xgiNtg+jw−∫LwidLtiRx2dxRpi2.The seepage resistances in the water area and CO2 area are given in Equations (64) and (65).
(64)Ww,iNtg+jw+1=8μwLciπRpi4+μw∫0LwiddxGx,(65)Wg,iNtg+jw+1=8μgxrgieπRpi4+μg∫LwidLtidxGx.At this time, the length of the crude oil zone isL−Lci−Lti−xrgie−xλiNtg+jw, and the resistance of the crude oil zone can be obtained by substituting Equation (49), and the seepage resistance of the mixed phase can be obtained by using Equation (48). By substituting the formula for seepage resistance in each zone into Equations (51) and (52), the water injection amount of the ith capillary and capillary bundle can be obtained. According to Equations (51)-(55), the length of the water zone and the CO2 zone can be obtained, and the method of obtaining the miscible zone is the same as in previous cases.


### 3.3.3. Fluid Distribution Pattern IX

At the beginning ofNtg+jg+1, if the calculated Lwid is greater than Lti, the water-CO2 interface has passed through the throat, and the length formed at the right end of throat isxwrie (e=1, e=e+1), xwrie=xwijg−Lci−∫0LtiRi2xdx/Rpi2..At the end ofNtg+jg, the actual length of the water area isxwijg=Lci+Lti+xwrie. xwijg is the equivalent length of water zone, the length of CO2 zone is xgiNig+jg, the length of miscible zone is xλiNig+jg, and the length of the crude oil zone is L−Lci−Lti−xλiNtg+jg−xgiNtg+jg−xwrie. Water-CO2 interfacial tension remains constant.At the beginning ofNtg+jg+1, the seepage resistance for each zone is as follows:
(66)Ww,iNtg+jg+1=8μwLci+xwrieπRI4+μw∫0LtidxGx,(67)Wg,iNtg+jg+1=8μgxgiNtg+jgπRi4,(68)Wλ,iNig+jg+1=8μ¯ogxλiNtg+jgπRi4,(69)Wo,iNtg+jg+1=8μoL−xwijg−xgiNtg+jg−xλiNtg+jgπRi4.Substituting Equations (66)-(69) into Equations (51)-(55), the length of the water zone and the CO2 zone can be obtained, and the parameters such as β and DE are substituted into Equation (40) to obtain the width of the miscible zone.


## 3.3.1. Fluid Distribution Pattern VII

At this time, the seepage resistance of the water zone is as follows:(50)Ww,ij=8μwxwij−1πRpi4,where xwij−1 is the length of the water zone in the capillary at the beginning of Ntg+j, cm, xwi0=0.The pressure difference between the ends of the capillary bundle isΔP+Pci, which is kept constant, and the flow rate of injected water is as follows:
(51)qw,ij=ΔP+PciWeq,iNtg+j,Pci=2σw−gcosθw−gRpi,where Weq,iNtg+j=Wwij+Wg,iNtg+j+Wλ,iNtg+j+Wo,iNtg+j, Wg,iNtg+j,Wλ,iNig+j,Wo,iNtg+j is the magnitude of seepage resistance in the CO2 zone, the miscible zone, and the crude oil zone at the beginning of Ntg+j.The total amount of water injected is as follows:(52)QwNtg+j=∑i=1nqw,iNtg+j.During this process, the water seepage velocity is the same as CO2.
(53)Wwij=qw,ijπRpi2=WgiNtg+j.According to the principle of material balance, due to the constant total injection volume, CO2 will continue to diffuse into the crude oil during the water injection process, the length of the CO2 zone and the crude oil zone will decrease, the width of the miscible zone will increase, and the calculation formula of the miscible zone will be for the distribution mode. For the first, second, third, and sixth fluid distribution pattern, the formula for calculating the length of the CO2 zone should be corrected to the following:
(54)xgiNtg+j=xgiNtg+j−1−ΔVmisijπRpi2,where ΔVmistjis the amount of increase in the amount of CO2 dissolved in the miscible zone, cm3; xgiNtg+j and xgiNtg+j−1are the equivalent lengths of the CO2 zone at the time of Ntg+j and Ntg+j−1, cm.
(55)ΔVmisij=VmisiNtg+j−VmisiNtg+j−1=12πRpi2xλiNtg+j−xλiNtg+j−1,where xλiNtg+j and xλiNtg+j−1 are the equivalent lengths of the miscible band at the time of Ntg+j and Ntg+j−1. Since the diffusion of CO2 in water is not considered, the length of water zone in the capillary is as follows:
(56)xwij=xwij−1+qwijΔtπRpi2.


## 3.3.2. Fluid Distribution Pattern VIII

Assuming at the beginning ofNtg+jw+1 moment (which is also at the end of Ntg+jw moment), the length of water zone is calculated as xwijw>Lci. It means that at the end of Ntg+jw, the injected water has entered the throat. The length needs to be corrected, and the actual water zone length isxwijw−1.
(57)xw,ijw=Lci+Lwid,(58)Rpi2xwijw−Lci=∫0LwidR2xdx,where xwijw is the equivalent length of the initial water zone at the time of Ntg+jw+1, cm. Lwid (d=1, d=d+1) is the extended length of the injected water in the throat.In the throat, the radius of the water-CO2 interface changes, so it will affect the capillary force. The capillary radius and capillary force at Lwid are as follows:
(59)RiLwid=RP+Rt2+Rp−Rt2cos2πLwidLt,(60)Pcid=2σw−gcosθ+γRix,(61)γ=tan−1dRxdx,(62)tanγ=dRxdx=RiLwid−RiLwid−1Lwid−Lwid−1,where RiLwi0=Rpi,Lwi0=0; γ is the correction value of the contact angle.At the beginning ofNtg+jw+1, the actual CO2 equivalent length is xgiNtg+jw, where the extension length at the right end of the throat is xrgie (e=1, e=e+1).
(63)xrgie=xgiNtg+jw−∫LwidLtiRx2dxRpi2.The seepage resistances in the water area and CO2 area are given in Equations (64) and (65).
(64)Ww,iNtg+jw+1=8μwLciπRpi4+μw∫0LwiddxGx,(65)Wg,iNtg+jw+1=8μgxrgieπRpi4+μg∫LwidLtidxGx.At this time, the length of the crude oil zone isL−Lci−Lti−xrgie−xλiNtg+jw, and the resistance of the crude oil zone can be obtained by substituting Equation (49), and the seepage resistance of the mixed phase can be obtained by using Equation (48). By substituting the formula for seepage resistance in each zone into Equations (51) and (52), the water injection amount of the ith capillary and capillary bundle can be obtained. According to Equations (51)-(55), the length of the water zone and the CO2 zone can be obtained, and the method of obtaining the miscible zone is the same as in previous cases.


## 3.3.3. Fluid Distribution Pattern IX

At the beginning ofNtg+jg+1, if the calculated Lwid is greater than Lti, the water-CO2 interface has passed through the throat, and the length formed at the right end of throat isxwrie (e=1, e=e+1), xwrie=xwijg−Lci−∫0LtiRi2xdx/Rpi2..At the end ofNtg+jg, the actual length of the water area isxwijg=Lci+Lti+xwrie. xwijg is the equivalent length of water zone, the length of CO2 zone is xgiNig+jg, the length of miscible zone is xλiNig+jg, and the length of the crude oil zone is L−Lci−Lti−xλiNtg+jg−xgiNtg+jg−xwrie. Water-CO2 interfacial tension remains constant.At the beginning ofNtg+jg+1, the seepage resistance for each zone is as follows:
(66)Ww,iNtg+jg+1=8μwLci+xwrieπRI4+μw∫0LtidxGx,(67)Wg,iNtg+jg+1=8μgxgiNtg+jgπRi4,(68)Wλ,iNig+jg+1=8μ¯ogxλiNtg+jgπRi4,(69)Wo,iNtg+jg+1=8μoL−xwijg−xgiNtg+jg−xλiNtg+jgπRi4.Substituting Equations (66)-(69) into Equations (51)-(55), the length of the water zone and the CO2 zone can be obtained, and the parameters such as β and DE are substituted into Equation (40) to obtain the width of the miscible zone.


## 4. Sensitivity Analysis

### 4.1. Model Solving Process and Basic Parameter Settings

According to the above descriptions, the solution process can be divided into two steps, CO2 injection and water injection, and the flow chart of solution is shown in Figure 4.Figure 4
Flow diagram of solution.During water-alternating-gas flooding process, the injection capacity is defined as the ratio of injected CO2/water rate to the pressure difference across the capillary bundle:
(70)I=Qw,gΔPw,g,where I is the injection capacity, cm3/(MPa·s); Qw,g is the CO2/water injection flow rate, cm3/s; ΔPw,g is the pressure difference across the capillary beam, MPa.For the convenience of analysis, it is stipulated that during the injection of CO2 and water injection, the pressure difference across the capillary bundle is equal. To quantitatively characterize the ability of the capillary tube to flow, the equivalent permeability is introduced, and the expression is as follows:
(71)k=L8∑i=1nRpi21/Rpi4+∫0Lti1/Ri4xdx.According to the variables used in the model derivation process, the pore throat size of the reservoir in low-permeability reservoirs and the characteristics of fluid properties of water-alternating-CO2 injection are investigated [35], and the basic parameters of the model are set, as shown in Table 2.Table 2
Parameters for CO2 water-alternating-gas model.
ParametersValueParametersValueCapillary radius (μm)5~15Capillary length (cm)30Water/CO2 interfacial tension (mN/m)20Water/CO2 contact angle (degree)60Crude oil viscosity (mPa·s)5CO2 viscosity (mPa·s)0.05Water viscosity (mPa·s)2Kw (cm)0.001Kμ(cm/(Pa∙s))0.001CO2 injection time (s)5Dm (cm2/s)1×10−5Time interval (s)0.1Injection time (s)5Pore throat ratio3Pressure difference at both ends of the capillary bundle (MPa)1Distance between throat and inlet end (cm)2~3Throat length (cm)1Number of capillaries11


### 4.2. Injection Capacity Analysis

To better reflect the change law of injectivity of CO2, mixed-phase slug, and injection capacity, the throat is set closer to the injection end. Then, the influences of viscosity, pore throat ratio, CO2 slug size, and equivalent permeability on the variance of CO2-alternating-water capacity have been studied.


#### 4.2.1. Crude Oil Viscosity

Assuming the crude oil viscosities are 5 mPa·s, 6 mPa·s, and 7 mPa·s, and simultaneously keep all other parameters constant, the changing law of injectivity with time is plotted in Figure5.Figure 5
Injectivity change with time for different oil viscosity.As it can be seen from Figure5, the crude oil viscosity increases, and the viscous resistance that the fluid flow needs to overcome increases, thus causing the injectivity to decrease as the crude oil viscosity increases. After the capillary beam is transferred to water, the high-viscosity crude oil is driven out. Then, the total seepage resistance of the fluid in the capillary bundle is reduced, causing the increase of injectivity. However, when the water-flooding front enters the throat, the low-viscosity CO2 is replaced. Then, the seepage resistance is increased, causing a decrease of injectivity. As the fluid percolation velocity decreases with the increase of the crude oil viscosity, the time that the front of the water flooding enters the throat continuously is delayed with the increase of the crude oil viscosity. When the water-gas interface, that is, the front edge of the water drive passes through the throat, the throat is occupied by water, the displacement will tend to be stable, and the water injection capacity will continue to increase.


#### 4.2.2. Pore Throat Ratio

Since the capillary radius has been set to characterize the pore size, the pore throat ratio is defined as the ratio of the capillary radius to the radius at the narrowest point of the throat. Set them to 2.5, 3, and 3.5, respectively, and the injection capacity changes with time are shown in Figure6.Figure 6
Injectivity change with time for different pore-throat ratio.When the capillary radius is constant (the pore radius is constant), the throat radius decreases with the increase of the pore throat ratio. Because the throat size has a great influence on the seepage resistance, the CO2 water-gas alternate injection capacity decreases with the increase of throat ratio. The throat is closer to the injection end, and after the water is injected, the injectivity is increased because the crude oil in the capillary bundle is replaced by the injected water. Similar to Figure 5, the front edge of the water drive enters the throat, and the injectivity is rapidly reduced; after the front edge passes through the throat, the injection capacity well increase slowly. Because the seepage resistance generated by the fluid in the throat is large, after the water flooding front passes through throat, the rising rate of water injectivity will decrease with the increase of pore throat ratio.


#### 4.2.3. CO2 Slug Size

Because CO2 is injected with constant pressure, the size of CO2 slug cannot be changed by changing the CO2 injection rate. However, CO2 slugs with different sizes can be equivalently injected by setting different injection times. According to this treatment method, CO2 injection time is set to be 4 s, 5 s, and 6 s, and then, different CO2 slug size is obtained. After CO2 injection, the changes of injectivity with time are shown in Figure 7.Figure 7
Injectivity change with time for different CO2 slug size.The larger the CO2 injection time is, the longer the CO2 slug is formed, and the smaller the total seepage resistance of the fluid in the capillary bundle is, so the injection capacity increases with the injection time. The throat is closer to the injection end, and the injection capacity increases linearly before the front of the water drive enters the throat. When the front edge of the water drive enters the throat, the injectivity decreases sharply. When the water drive front passes through the throat, the injectivity will increase slowly.


#### 4.2.4. Equivalent Permeability

Since the pore throat ratio between the capillary and the throat is constant, the capillary radius distribution range in the capillary bundle is changed, and the equivalent permeability under different capillary bundles can be obtained by Equation (71), which is 10.6×10−3μm2, 18.5×10−3μm2, and 28.5×10−3μm2. The injectivity is calculated at different permeability, and the resulting injection capacity changes with time are shown in Figure 8.Figure 8
Injectivity change with time for different capillary tube radius.As can be seen from Figure8, the smaller the equivalent permeability (the smaller the range of the capillary bundle radius) is given, the smaller the injectivity will be. The reason is that for crude oil with the same viscosity, the smaller the equivalent permeability, the greater the resistance to be overcome by the fluid flow. Under the same injection pressure difference, the smaller the injection flow rate, the lower the injection capacity. This change law indicates that for low-permeability reservoirs, the injection capacity is lower due to the smaller pore throat, and the required injection pressure difference is greater at the same injection amount compared with the high-permeability reservoir. Under the same injection amount, the required water injection pressure difference is larger. If the fluid is injected with constant pressure, it is hard to inject water, which has an impact on the oil recovery for low-permeability reservoirs. Comparing Figure 5 to Figure 8, it can be found that the change of equivalent permeability has the greatest influence on the injection capacity, which also indicates that in a practical pilot test for CO2-alternating-water injection, the influence of permeability on injectivity should be fully considered.


## 4.1. Model Solving Process and Basic Parameter Settings

According to the above descriptions, the solution process can be divided into two steps, CO2 injection and water injection, and the flow chart of solution is shown in Figure 4.Figure 4
Flow diagram of solution.During water-alternating-gas flooding process, the injection capacity is defined as the ratio of injected CO2/water rate to the pressure difference across the capillary bundle:
(70)I=Qw,gΔPw,g,where I is the injection capacity, cm3/(MPa·s); Qw,g is the CO2/water injection flow rate, cm3/s; ΔPw,g is the pressure difference across the capillary beam, MPa.For the convenience of analysis, it is stipulated that during the injection of CO2 and water injection, the pressure difference across the capillary bundle is equal. To quantitatively characterize the ability of the capillary tube to flow, the equivalent permeability is introduced, and the expression is as follows:
(71)k=L8∑i=1nRpi21/Rpi4+∫0Lti1/Ri4xdx.According to the variables used in the model derivation process, the pore throat size of the reservoir in low-permeability reservoirs and the characteristics of fluid properties of water-alternating-CO2 injection are investigated [35], and the basic parameters of the model are set, as shown in Table 2.Table 2
Parameters for CO2 water-alternating-gas model.
ParametersValueParametersValueCapillary radius (μm)5~15Capillary length (cm)30Water/CO2 interfacial tension (mN/m)20Water/CO2 contact angle (degree)60Crude oil viscosity (mPa·s)5CO2 viscosity (mPa·s)0.05Water viscosity (mPa·s)2Kw (cm)0.001Kμ(cm/(Pa∙s))0.001CO2 injection time (s)5Dm (cm2/s)1×10−5Time interval (s)0.1Injection time (s)5Pore throat ratio3Pressure difference at both ends of the capillary bundle (MPa)1Distance between throat and inlet end (cm)2~3Throat length (cm)1Number of capillaries11


## 4.2. Injection Capacity Analysis

To better reflect the change law of injectivity of CO2, mixed-phase slug, and injection capacity, the throat is set closer to the injection end. Then, the influences of viscosity, pore throat ratio, CO2 slug size, and equivalent permeability on the variance of CO2-alternating-water capacity have been studied.


### 4.2.1. Crude Oil Viscosity

Assuming the crude oil viscosities are 5 mPa·s, 6 mPa·s, and 7 mPa·s, and simultaneously keep all other parameters constant, the changing law of injectivity with time is plotted in Figure5.Figure 5
Injectivity change with time for different oil viscosity.As it can be seen from Figure5, the crude oil viscosity increases, and the viscous resistance that the fluid flow needs to overcome increases, thus causing the injectivity to decrease as the crude oil viscosity increases. After the capillary beam is transferred to water, the high-viscosity crude oil is driven out. Then, the total seepage resistance of the fluid in the capillary bundle is reduced, causing the increase of injectivity. However, when the water-flooding front enters the throat, the low-viscosity CO2 is replaced. Then, the seepage resistance is increased, causing a decrease of injectivity. As the fluid percolation velocity decreases with the increase of the crude oil viscosity, the time that the front of the water flooding enters the throat continuously is delayed with the increase of the crude oil viscosity. When the water-gas interface, that is, the front edge of the water drive passes through the throat, the throat is occupied by water, the displacement will tend to be stable, and the water injection capacity will continue to increase.


### 4.2.2. Pore Throat Ratio

Since the capillary radius has been set to characterize the pore size, the pore throat ratio is defined as the ratio of the capillary radius to the radius at the narrowest point of the throat. Set them to 2.5, 3, and 3.5, respectively, and the injection capacity changes with time are shown in Figure6.Figure 6
Injectivity change with time for different pore-throat ratio.When the capillary radius is constant (the pore radius is constant), the throat radius decreases with the increase of the pore throat ratio. Because the throat size has a great influence on the seepage resistance, the CO2 water-gas alternate injection capacity decreases with the increase of throat ratio. The throat is closer to the injection end, and after the water is injected, the injectivity is increased because the crude oil in the capillary bundle is replaced by the injected water. Similar to Figure 5, the front edge of the water drive enters the throat, and the injectivity is rapidly reduced; after the front edge passes through the throat, the injection capacity well increase slowly. Because the seepage resistance generated by the fluid in the throat is large, after the water flooding front passes through throat, the rising rate of water injectivity will decrease with the increase of pore throat ratio.


### 4.2.3. CO2 Slug Size

Because CO2 is injected with constant pressure, the size of CO2 slug cannot be changed by changing the CO2 injection rate. However, CO2 slugs with different sizes can be equivalently injected by setting different injection times. According to this treatment method, CO2 injection time is set to be 4 s, 5 s, and 6 s, and then, different CO2 slug size is obtained. After CO2 injection, the changes of injectivity with time are shown in Figure 7.Figure 7
Injectivity change with time for different CO2 slug size.The larger the CO2 injection time is, the longer the CO2 slug is formed, and the smaller the total seepage resistance of the fluid in the capillary bundle is, so the injection capacity increases with the injection time. The throat is closer to the injection end, and the injection capacity increases linearly before the front of the water drive enters the throat. When the front edge of the water drive enters the throat, the injectivity decreases sharply. When the water drive front passes through the throat, the injectivity will increase slowly.


### 4.2.4. Equivalent Permeability

Since the pore throat ratio between the capillary and the throat is constant, the capillary radius distribution range in the capillary bundle is changed, and the equivalent permeability under different capillary bundles can be obtained by Equation (71), which is 10.6×10−3μm2, 18.5×10−3μm2, and 28.5×10−3μm2. The injectivity is calculated at different permeability, and the resulting injection capacity changes with time are shown in Figure 8.Figure 8
Injectivity change with time for different capillary tube radius.As can be seen from Figure8, the smaller the equivalent permeability (the smaller the range of the capillary bundle radius) is given, the smaller the injectivity will be. The reason is that for crude oil with the same viscosity, the smaller the equivalent permeability, the greater the resistance to be overcome by the fluid flow. Under the same injection pressure difference, the smaller the injection flow rate, the lower the injection capacity. This change law indicates that for low-permeability reservoirs, the injection capacity is lower due to the smaller pore throat, and the required injection pressure difference is greater at the same injection amount compared with the high-permeability reservoir. Under the same injection amount, the required water injection pressure difference is larger. If the fluid is injected with constant pressure, it is hard to inject water, which has an impact on the oil recovery for low-permeability reservoirs. Comparing Figure 5 to Figure 8, it can be found that the change of equivalent permeability has the greatest influence on the injection capacity, which also indicates that in a practical pilot test for CO2-alternating-water injection, the influence of permeability on injectivity should be fully considered.


## 4.2.1. Crude Oil Viscosity

Assuming the crude oil viscosities are 5 mPa·s, 6 mPa·s, and 7 mPa·s, and simultaneously keep all other parameters constant, the changing law of injectivity with time is plotted in Figure5.Figure 5
Injectivity change with time for different oil viscosity.As it can be seen from Figure5, the crude oil viscosity increases, and the viscous resistance that the fluid flow needs to overcome increases, thus causing the injectivity to decrease as the crude oil viscosity increases. After the capillary beam is transferred to water, the high-viscosity crude oil is driven out. Then, the total seepage resistance of the fluid in the capillary bundle is reduced, causing the increase of injectivity. However, when the water-flooding front enters the throat, the low-viscosity CO2 is replaced. Then, the seepage resistance is increased, causing a decrease of injectivity. As the fluid percolation velocity decreases with the increase of the crude oil viscosity, the time that the front of the water flooding enters the throat continuously is delayed with the increase of the crude oil viscosity. When the water-gas interface, that is, the front edge of the water drive passes through the throat, the throat is occupied by water, the displacement will tend to be stable, and the water injection capacity will continue to increase.


## 4.2.2. Pore Throat Ratio

Since the capillary radius has been set to characterize the pore size, the pore throat ratio is defined as the ratio of the capillary radius to the radius at the narrowest point of the throat. Set them to 2.5, 3, and 3.5, respectively, and the injection capacity changes with time are shown in Figure6.Figure 6
Injectivity change with time for different pore-throat ratio.When the capillary radius is constant (the pore radius is constant), the throat radius decreases with the increase of the pore throat ratio. Because the throat size has a great influence on the seepage resistance, the CO2 water-gas alternate injection capacity decreases with the increase of throat ratio. The throat is closer to the injection end, and after the water is injected, the injectivity is increased because the crude oil in the capillary bundle is replaced by the injected water. Similar to Figure 5, the front edge of the water drive enters the throat, and the injectivity is rapidly reduced; after the front edge passes through the throat, the injection capacity well increase slowly. Because the seepage resistance generated by the fluid in the throat is large, after the water flooding front passes through throat, the rising rate of water injectivity will decrease with the increase of pore throat ratio.


## 4.2.3. CO2 Slug Size

Because CO2 is injected with constant pressure, the size of CO2 slug cannot be changed by changing the CO2 injection rate. However, CO2 slugs with different sizes can be equivalently injected by setting different injection times. According to this treatment method, CO2 injection time is set to be 4 s, 5 s, and 6 s, and then, different CO2 slug size is obtained. After CO2 injection, the changes of injectivity with time are shown in Figure 7.Figure 7
Injectivity change with time for different CO2 slug size.The larger the CO2 injection time is, the longer the CO2 slug is formed, and the smaller the total seepage resistance of the fluid in the capillary bundle is, so the injection capacity increases with the injection time. The throat is closer to the injection end, and the injection capacity increases linearly before the front of the water drive enters the throat. When the front edge of the water drive enters the throat, the injectivity decreases sharply. When the water drive front passes through the throat, the injectivity will increase slowly.


## 4.2.4. Equivalent Permeability

Since the pore throat ratio between the capillary and the throat is constant, the capillary radius distribution range in the capillary bundle is changed, and the equivalent permeability under different capillary bundles can be obtained by Equation (71), which is 10.6×10−3μm2, 18.5×10−3μm2, and 28.5×10−3μm2. The injectivity is calculated at different permeability, and the resulting injection capacity changes with time are shown in Figure 8.Figure 8
Injectivity change with time for different capillary tube radius.As can be seen from Figure8, the smaller the equivalent permeability (the smaller the range of the capillary bundle radius) is given, the smaller the injectivity will be. The reason is that for crude oil with the same viscosity, the smaller the equivalent permeability, the greater the resistance to be overcome by the fluid flow. Under the same injection pressure difference, the smaller the injection flow rate, the lower the injection capacity. This change law indicates that for low-permeability reservoirs, the injection capacity is lower due to the smaller pore throat, and the required injection pressure difference is greater at the same injection amount compared with the high-permeability reservoir. Under the same injection amount, the required water injection pressure difference is larger. If the fluid is injected with constant pressure, it is hard to inject water, which has an impact on the oil recovery for low-permeability reservoirs. Comparing Figure 5 to Figure 8, it can be found that the change of equivalent permeability has the greatest influence on the injection capacity, which also indicates that in a practical pilot test for CO2-alternating-water injection, the influence of permeability on injectivity should be fully considered.


## 5. Conclusions

(1)
Considering the seepage, mass transfer, and CO2 diffusion during CO2 flooding, a mathematical model of convection-diffusion is established and solved. The relationship between the width of the mixed-phase zone and the injection time is obtained(2)
According to the difference of distribution forms for the unswept oil region, mixed-phase zone, CO2 region, and water region in a capillary tube, six possible patterns are defined during CO2 displacement, and three distribution patterns are described during water flooding. Through the introduction of the equivalent resistance model, the injectivity for each possible distribution pattern is evaluated during water-alternating-gas displacement(3)
Through parameter analysis, the injectivity decreases with the increase of crude oil viscosity and pore throat ratio. Oppositely, the injectivity increases with the equivalent permeability and CO2 slug size. The equivalent permeability has the greatest influence on injection capacity compared with other factors; therefore, in practice, the influence of permeability on injectivity should be carefully considered



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*Source: 8836287-2020-11-29.xml*