A compositional model for CO_{2} flooding including CO_{2} equilibria between water and oil using the Peng–Robinson equation of state with the Wong–Sandler mixing rule
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Abstract
This paper presents a three-dimensional, three-phase compositional model considering CO_{2} phase equilibrium between water and oil. In this model, CO_{2} is mutually soluble in aqueous and hydrocarbon phases, while other components, except water, exist in hydrocarbon phase. The Peng–Robinson (PR) equation of state and the Wong–Sandler mixing rule with non-random two-liquid parameters are used to calculate CO_{2} fugacity in the aqueous phase. One-dimensional and three-dimensional CO_{2} flooding examples show that a significant amount of injected CO_{2} is dissolved in water. Our simulation shows 7% of injected CO_{2} can be dissolved in the aqueous phase, which delays oil recovery by 4%. The gas rate predicted by the model is smaller than the conventional model as long as water is undersaturated by CO_{2}, which can be considered as “lost” in the aqueous phase. The model also predicts that the delayed oil can be recovered after the gas breakthrough, indicating that delayed oil is hard to recover in field applications. A three-dimensional example reveals that a highly stratified reservoir causes uneven displacement and serious CO_{2} breakthrough. If mobility control measures like water alternating gas are undertaken, the solubility effects will be more pronounced than this example.
Keywords
CO_{2} flooding Wong–Sandler mixing rule Equation of state Numerical simulation CO_{2} solubilityList of symbols
- \(\underline{A}\)
Molar Helmholtz free energy
- A
Area of surface between two grid blocks
- B
Second virial coefficient
- \(B_{\text{w}}^{*}\)
Formation volume factor of a brine without solution gas
- B_{w}
FVF of a brine at saturated condition
- F
Arbitrary function
- L
Distance between two grid blocks
- K
Permeability, mD
- K_{r}
Relative permeability
- K_{rw}
Relative permeability of water
- K_{rgcw}
Relative permeability of gas at connate water saturation
- K_{rocw}
Relative permeability of oil at connate water saturation
- Q
Quadratic sum of second virial coefficients
- R_{sw}
Gas solubility in the aqueous phase (scf/STB)
- R
Gas constant (J K^{−1} mol^{−1})
- P
Pressure (Pa)
- S_{o}, S_{g}, S_{w}
Oil, gas, and water saturation, fraction
- S_{wc}
Connate water saturation
- S_{org}
Residual oil saturation to gas
- S_{gc}
Critical gas saturaion
- S_{gr}
Residual gas saturation
- T
Transmissibility (mD m)
- T_{F}
Temperature in Fahrenheit (°F)
- T_{cels}
Temperature in Celsius (°C)
- \(\underline{V}\)
Molar volume (m^{3}/mol)
- a
Equation of state “energy” parameter
- b
Equation of state “excluded volume” parameter
- \(c_{\text{w}}^{ *}\)
Compressibility of a brine without solution gas (psi^{−1})
- c_{w}
Compressibility of a brine without solution gas (psi^{−1})
- f_{i}
Fugacity of component i at zero salinity (Pa)
- f_{i,s}
Fugacity of component i in brine (salt solution) (Pa)
- f_{i,o}
Fugacity of component i in oil (Pa)
- f_{i,g}
Fugacity of component i in gas (Pa)
- \(f_{{{\text{CO}}_{ 2} , {\text{o}}}}\)
CO_{2} fugacity in oil (Pa)
- \(f_{{{\text{CO}}_{ 2} , {\text{g}}}}\)
CO_{2} fugacity in gas (Pa)
- \(f_{{{\text{CO}}_{2} ,{\text{w}}}}\)
CO_{2} fugacity in water (Pa)
- g
Local composition factor for the NRTL model
- k_{ij}
Binary interaction coefficient between components i and j
- k_{is}
Salting-out coefficient for component i
- \(k_{{{\text{CO}}_{ 2} , {\text{s}}}}\)
Salting-out coefficient for component CO_{2}
- m_{s}
Molality of the dissolved salt (mol/kg)
- q_{w}, q_{i}
Water well rate and hydrocarbon component i well rate (mol/s)
- p_{c,ow}, p_{c,go}
Oil–water and gas–oil capillary pressure, respectively (Pa)
- p_{o}, p_{g}, p_{w}
Oil, gas, and water pressure, respectively (Pa)
- w_{s}
Weight fraction of NaCl (fraction)
- x_{i}, y_{i}
Mole fraction of component i in the oil and gas phases, fraction
- \(x_{{{\text{w}}\,{\text{CO}}_{ 2} }}\)
Mole fraction of CO_{2} in the aqueous phase (fraction)
- \(x_{{{\text{wH}}_{ 2} {\text{O}}}}\)
Mole fraction of H_{2}O in the aqueous phase (fraction)
- Δ
Difference operator
- \(\delta_{{i,{\text{CO}}_{ 2} }}\)
Kronecker delta
- \(\lambda_{\text{o}} ,\;\lambda_{\text{g}} ,\;\lambda_{\text{w}}\)
Oil, gas, and water mobilities (1/cP)
- \(\varPhi_{\text{o}} ,\;\varPhi_{\text{g}} ,\;\varPhi_{\text{w}}\)
Oil, gas, and water potentials (Pa)
- \(\mu\)
Viscosity (cP)
- \(\alpha\)
NRTL model parameter
- \(\tau\)
NRTL model binary interaction parameter
- \(\varphi\)
Fugacity coefficient
- \(\gamma\)
Activity coefficient
- \(\phi\)
Porosity (fraction)
- o, g, w
Oil, gas, and aqueous phases, respectively
- i
Component index
- p
Phase (o, g, w)
- *
Simulation results of the conventional model
1 Introduction
Modeling transient flow of reservoir fluids with large variations in compositions requires a compositional model (Ju et al. 2012; Cho et al. 2018). It has become mainstream to simulate multiphase flow such as miscible gas injection and depletion of volatile oil/gas-condensate reservoirs. In most published models, for example, Fussell and Fussell (1979), Coats (1980), Young and Stephenson (1983), and Chien et al. (1985), water is treated as an independent component where hydrocarbon components are not allowed to dissolve. This assumption is appropriate for gas injection where the components are hard to dissolve in the aqueous phase. CO_{2} flooding, however, is not suited for this assumption because CO_{2} solubility in water is much higher than that of hydrocarbons, whose effect cannot be neglected in the simulation process. This is especially true when CO_{2} is injected into previously waterflooded reservoirs or into tight oil reservoirs where connate water saturation is up to 50%. Enick and Klara (1992) and Chang et al. (1998) demonstrated that the CO_{2} dissolved in the formation brine accounts for a significant fraction of the total amount of CO_{2} injected into the reservoir and the CO_{2} solubility has a substantially adverse effect on the ultimate recovery. Therefore, a reliable and efficient compositional simulator including CO_{2} solubility in water is needed to achieve more accurate simulation results.
The first challenge to develop such a simulator is how to model the phase behavior in the aqueous phase. Many researchers have experimentally studied the binary CO_{2}/water system (King et al. 1992; Valtz et al. 2004; Guo et al. 2014) and proposed numerous approaches to model the behavior (Pedersen et al. 2001; Spycher et al. 2003). The traditional fugacity approach that uses the cubic equation of state (EOS) with the van der Waals (vdW) mixing rule correlates the phase behavior of hydrocarbon mixtures accurately as long as appropriate binary interaction parameters are selected (Zhao and Lvov 2016). However, it is insufficient to obtain reliable results for a mixture containing strongly polar components like water. Over the past three decades, much effort has been devoted to modifying or replacing the vdW one-fluid mixing rule for the challenging vapor–liquid equilibrium calculation (Zhao and Lvov 2016).
Among the modern approaches presented in the literature, a method of the type “EOS + excess Gibbs free energy” (EOS/G^{ex}) is the most adequate for modeling mixtures with highly asymmetric components. Mixing rules proposed by Huron and Vidal (1979) and Wong and Sandler (1992), belonging to the EOS/G^{ex} type, have been extensively used and applied to highly challenging phase equilibria. Compared with the Huron–Vidal mixing rule, the Wong–Sandler (WS) mixing rule satisfies the quadratic mole fraction dependence of the second virial coefficient and predicts the same excess Helmholtz energy at infinite pressure as a function of composition as that obtained from a selected activity coefficient model (Zhao and Lvov 2016). Many studies show the combination of WS mixing rule with non-random, two-liquid (NRTL) parameters gives the best results for water-containing polar mixtures compared to other mixing rules coupling cubic EOS (Valderrama 2003). Jaubert and Mutelet (2004) and Jaubert et al. (2010) proposed a group contribution-based thermodynamic model (PPR78 EOS) which combines, at a constant packing fraction, the Peng–Robinson (PR) EOS and a van Laar-type G^{ex} model. Their article demonstrates that using classical mixing rules, the PPR78 model is able to estimate the temperature-dependent K_{ij} (the binary interacting coefficient) for any mixtures containing alkanes, aromatics, naphthenes, CO_{2}, N_{2}, H_{2}S, and mercaptans. The innovative part of their work is to establish a predictive model that is able to estimate the interactions from mere knowledge of the structure of molecules within the petroleum blend. They proved that the PPR78 model can reliably predict the vapor–liquid equilibrium (VLE) of very asymmetric systems, which points out a new trend to predict the phase behavior of polar–nonpolar systems.
In addition to the EOS approach, Li and Nghiem (1986) used Henry’s law to estimate CO_{2} solubility in distilled water and used the scaled particle theory to take into account the presence of salt in the aqueous phase. The fugacity coefficients of light components that are considered soluble in the aqueous phase (e.g., methane, ethane, propane, and CO_{2}) can be derived from Henry’s constant, and for hydrocarbon phase, they are calculated by the conventional cubic EOS with the vdW mixing rule. A time-consuming three-phase flash calculation is accomplished to obtain the phase equilibrium of the water–oil–gas system. Enick and Klara (1990) used the Krichevsky–llinskaya equation to correlate the solubility of CO_{2} in water, and the decreased solubility of CO_{2} in brine was accounted for empirically by a single factor correlated with the weight percent of dissolved solids. Chang et al. (1998) also proposed an empirical correlation for the solubility of CO_{2} in distilled water as a function of temperature. The solubility in distilled water can be adjusted further for the effect of salinity to obtain the solubility of CO_{2} in brine. Apart from this simple correlation, Chang et al. proposed an isothermal, three-dimensional composition model with both fully implicit and implicit pressure explicit saturation formulations. The innovative point of their work is that CO_{2} fugacity coefficients in the aqueous phase are computed internally from the correlation and the equal-fugacity constraint of CO_{2} for aqueous and hydrocarbon phases is introduced to solve the aqueous composition. Yan and Stenby (2009, 2010) used the PR EOS modified by Søreide and Whitson to describe the phase equilibrium between CO_{2} and brine. A one-dimensional slim tube simulator combined with a multiphase flash subroutine was proposed to model CO_{2} flooding considering the influence of CO_{2} solubility, where the aqueous phase was treated as an inert phase or only dissolving CO_{2}.
To consider the CO_{2} solubility in water, the existing reservoir compositional simulation is either with the aid of Henry’s law or with empirical correlations. Although the EOS/G^{ex} approach is used widely in predicting the fluid phase equilibrium of polar components in process design, this model has never been integrated into reservoir simulation to fulfill the challenging simulation of CO_{2} flooding including CO_{2} equilibria between water and oil. In this work, we validate the EOS/G^{ex} model by reproducing the experimental PVT data of the binary CO_{2}–H_{2}O system and then provide formulations about how to integrate the EOS/G^{ex} model into the reservoir compositional simulation. Finally, we compare the simulation performance of our model with the existing compositional model. The simulation results of this study help to improve the accuracy of the numerical simulation of the oil recovery process involving CO_{2}, which will, in turn, improve the quality of the reservoir performance prediction and the reliability of the economic calculations.
2 General description of the model
The simulator described here is an isothermal, three-dimensional compositional model. Fully implicit formulations are presented which are able to treat water, oil, and gas flow through reservoirs of heterogeneous permeability and porosity. It is designed to model the compositional flow process considering the CO_{2} phase equilibrium between the aqueous phase and the hydrocarbon phase. This simulator does not model the three-hydrocarbon-phase phenomenon that has been observed for some CO_{2}/hydrocarbon systems.
The model consists of mass balance equations for water and n_{c} hydrocarbon components and associated constraint equations. Oil- and gas-phase densities and fugacities are calculated from the PR EOS, while the CO_{2} fugacity in the aqueous phase, adjusted by a salting-out coefficient, is calculated by PR EOS with WS mixing rule (we denote the model proposed in this paper as the PR–WS model). Oil and gas viscosities are calculated by the Lohrenz et al. (1964) method. The fluid flow is simulated with Darcy’s law, incorporating viscous, gravitational, and capillary forces.
Formulation assumptions are an instantaneous equilibrium between gas and oil phases in each grid, and only CO_{2} is considered mutually soluble in water and oil. One reason for this is that the CO_{2} solubility in water is significantly larger than other hydrocarbon components. Another reason is that when large quantities of CO_{2} are injected during CO_{2} flooding, it becomes the dominant component dissolved in water compared to other relatively highly soluble constituents like H_{2}S. This assumption is able to simplify the phase behavior calculation in the aqueous phase and reduce computation time greatly. Moreover, an equal-fugacity constraint of CO_{2} between water and oil is used to characterize partitioning CO_{2} in water and oil.
3 The PR–WS composition model
3.1 Reservoir model equations
Mass balance-type equations are used to describe the multicomponent multiphase flow in porous media. These equations can be divided into three parts (Young and Stephenson 1983): (1) mass balance equations describing component flow, (2) phase equilibrium relationships, and (3) constraint equations that require the phase saturation to sum to unity and the mole fraction in each phase to sum to unity.
For simplicity of development, the specific model assumptions are summarized as follows: (1) The dispersion and gravity forces are neglected, (2) only CO_{2} is considered mutually soluble in water and oil, and water has no mass exchange with the hydrocarbon, (3) the effect of gas (mainly CO_{2}) on the aqueous viscosity is not considered because it was found to be very small, and (4) aqueous phase properties, such as the formation volume factor, water compressibility, and water viscosity, are calculated by empirical correlations.
- (1)
The \(\;n_{\text{c}} + 1\) mass balance equations:
- (2)
The \(\;n_{\text{c}} + 1\) fugacity equations (for a three-phase block)
- (3)
Six constraint equations
Equations and unknown variables in the reservoir simulation model
Type | Number | |
---|---|---|
Equations | F _{ i} | n _{c} |
F _{w} | 1 | |
F_{eh}, F_{ew} | n_{c} + 1 | |
F_{pcow}, F_{pcgo} | 2 | |
F _{s} | 1 | |
F_{po}, F_{pg}, F_{pw} | 3 | |
Total | 2n_{c} + 8 | |
Variables | p _{p} | 3 |
S _{p} | 3 | |
x _{ i} | n _{c} | |
y _{ i} | n _{c} | |
\(x_{{{\text{w}}\,{\text{CO}}_{2} }}\), \(x_{{{\text{w}}\,{\text{H}}_{ 2} {\text{O}}}}\) | 2 | |
Total | 2n_{c} + 8 |
Primary and secondary variables in the case of the three-phase block
Type | Equation | Variable |
---|---|---|
Primary | F _{ i} | \(p_{\text{o}} ,S_{\text{o}} ,x_{i} \;\;(i = 1, \ldots ,n_{\text{c}} - 1)\) |
F _{w} | S _{g} | |
Secondary | F _{eh} | \(x_{i} ,y_{i} \;\;(i = 1, \ldots ,n_{\text{c}} - 1)\) |
F _{ew} | \(x_{{{\text{w}}\,{\text{CO}}_{2} }}\) |
Primary variable selection
Type | Variable | F _{ew} |
---|---|---|
Water–oil–gas | \(p_{\text{o}} ,S_{\text{o}} ,S_{\text{g}} ,x_{i} \;\;(i = 2, \ldots ,n_{\text{c}} - 1)\) | \(F_{\text{ew}} = f_{{{\text{CO}}_{ 2} , {\text{w}}}} - f_{{{\text{CO}}_{ 2} , {\text{o}}}} = 0\) |
Water–oil | \(p_{\text{o}} ,S_{\text{o}} ,x_{i} \;\;(i = 1, \ldots ,n_{\text{c}} - 1)\) | \(F_{\text{ew}} = f_{{{\text{CO}}_{ 2} , {\text{w}}}} - f_{{{\text{CO}}_{ 2} , {\text{o}}}} = 0\) |
Water–gas | \(p_{\text{o}} ,S_{\text{o}} ,y_{i} \;\;(i = 1, \ldots ,n_{\text{c}} - 1)\) | \(F_{\text{ew}} = f_{{{\text{CO}}_{ 2} , {\text{w}}}} - f_{{{\text{CO}}_{ 2} , {\text{g}}}} = 0\) |
3.2 Fugacity of CO_{2} in water
PR EOS with the WS mixing rule with the NRTL model is used to calculate CO_{2} fugacity and its derivative in water. “Appendix 1” gives the derivation of the WS mixing rule for PR EOS (Wong and Sandler 1992). The calculation procedure is: (1) A successive substitution method is used to apply a flash calculation for the binary CO_{2}/water system, (2) interaction parameters for the WS mixing rule at the specified reservoir temperature are evaluated by fitting experimental data using flashing results, and (3) fitting parameters and PR EOS with the WS mixing rule are used to generate CO_{2} fugacity and derivatives in reservoir simulation.
Interaction parameters of the WS mixing rule evaluated from fitting published literature data (Bamberger et al. 2000) for the binary CO_{2}–H_{2}O system
T, K | \(\tau_{ij}\) | \(\tau_{ji}\) | \(\alpha\) | \(k_{ij}\) | \(k_{ji}\) |
---|---|---|---|---|---|
323 | 4.3870 | 0.3930 | 0.1141 | 0.3073 | 0.3073 |
333 | 4.3570 | 0.4130 | 0.1120 | 0.3073 | 0.3073 |
353 | 4.1270 | 0.4530 | 0.1041 | 0.3073 | 0.3073 |
3.3 Aqueous phase properties
Aqueous phase properties, like formation volume factor (B_{w}), water compressibility (c_{w}), and water viscosity, can be simply correlated with the methods displayed in “Appendix 3.” Initially, B_{w} and c_{w} should be evaluated at the CO_{2} saturated condition. For an undersaturated condition, linear interpolation is suggested by Chang et al. (1998). However, we found that aqueous phase properties are virtually proportional to the CO_{2} solubility. Hence, we replaced R_{sw} in Eqs. 35 and 37 with the mole fraction of CO_{2} in water to avoid cumbersome CO_{2} solubility prediction. This assumption introduces trivial error for the undersaturated condition because no more than a few percent of CO_{2} is in the aqueous phase. The effect of gas on aqueous viscosity is not considered because it was found to be very small (Whitson and Brulé 2000).
4 Simulation results
The simulation described here includes one- and three-dimensional CO_{2} flooding problems. For each problem, the PR–WS model is compared to the conventional model where the solubility of CO_{2} in water is ignored. Water is present but is immobile in all calculations to have a better understanding of phase equilibria of CO_{2}. The CO_{2}/methane/butane/decane system is used for the hydrocarbon content. The reservoir temperature is 333 K for all calculations, and interaction parameters for the WS mixing rule are taken from the evaluated data in Table 4 at 333 K. The capillary force, dispersion, and gravity are neglected for all simulations.
4.1 One-dimensional CO_{2} flooding
Model data of the one-dimensional problem
Item | Value |
---|---|
Reservoir length, m | 150 |
Reservoir width, m | 60 |
Reservoir thickness, m | 30 |
Permeability, mD | 20 |
Porosity | 0.2 |
Grid blocks in the x, y, z directions | 20 × 1 × 1 |
Capillary pressure | 0 |
Water compressibility c_{w}, MPa^{−1} | 0 |
Compressibility, MPa^{−1} | 0 |
Relative permeability data | |
S_{wc} | 0.2 |
S_{org} | 0.2 |
S_{gc} | 0 |
S_{gr} | 0.15 |
K_{rocw} | 1.0 |
K_{rgcw} | 1.0 |
K_{rw} | 0 |
Water viscosity, cP | 0.5 (constant) |
Initial pressure, MPa | 14 |
Reservoir temperature, K | 333 |
Initial oil composition (CO_{2}, C_{1}, C_{4}, C_{10}), mol% | 0.05, 0.15, 0.2, 0.6 |
Initial saturation (S_{w}, S_{o}, S_{g}) | |
run1 | 0.2, 0.8, 0.0 |
run2 | 0.4, 0.6, 0.0 |
run3 | 0.6, 0.4, 0.0 |
The standard conditions | |
Pressure, MPa | 0.101325 |
Temperature, K | 288.71 |
WS mixing rule parameters | From Table 4 at 333 K |
The oil and gas rate in run1 calculated by the PR–WS model is slightly lower than the conventional model because there is not much water present in the reservoir (S_{w} = 0.2). Run1 does not reach its turning point on account of too short production time or too small CO_{2} injection rate. ΔRF_{max} is at the end of the simulation with a value of 0.73%. Run2 displays a significant difference between the oil and gas rates of the PR–WS and conventional models. The turning point is around 1550 days. ΔRF_{max} locates at the turning point with a value of 1.93%, which shows around 2% of oil recovery is delayed. The phenomenon of delaying is enlarged in run3 when water saturation is increased to 0.6. This simulation shows that 4.11% of recovery is delayed, and the turning point is advanced to 1050 days. We observe that for a constant CO_{2} injection rate, the amount of oil present in the reservoir is a key factor which influences the time of the turning point, while the amount of water is a key factor which affects the value of ΔRF_{max}. Finally, the gas rate of the PR–WS model in the three runs is always smaller than the conventional model because a significant portion of injected CO_{2} is dissolved in water, which can be considered as “lost” in the aqueous phase.
4.2 Three-dimensional CO_{2} flooding
Model data of three-dimensional problem
Item | Value |
---|---|
Permeability of three layers k_{1}, k_{2}, k_{3}, mD | 500, 50, 200 |
Porosity | 0.3 |
Grid blocks in the x, y, z directions | 10 × 10 × 3 |
Dimensions (x, y, z), m | x = 305, y = 305, z_{1} = 6.10, z_{2} = 9.14, z_{3} = 15.24 |
Capillary pressure | 0 |
Water compressibility c_{w}, MPa^{−1} | Correlated with “Appendix 3” |
Compressibility, MPa^{−1} | 4.35 × 10^{−4} |
Relative permeability data | |
S_{wc} | 0.2 |
S_{org} | 0.2 |
S_{gc} | 0 |
S_{gr} | 0.15 |
K_{rocw} | 1.0 |
K_{rgcw} | 1.0 |
K_{rw} | 0 |
Water viscosity, cP | Correlated with “Appendix 3” |
Water formation volume factor | Correlated with “Appendix 3” |
Initial pressure, MPa | 14 |
Reservoir temperature, K | 333 |
Initial oil composition (CO_{2}, C_{1}, C_{4}, C_{10}), mol% | 0.05, 0.15, 0.2, 0.6 |
Initial saturation (S_{w}, S_{o}, S_{g}) | 0.4, 0.6, 0.0 |
The standard conditions | |
Pressure, MPa | 0.101325 |
Temperature, K | 288.71 |
WS mixing rule parameters | From Table 4 at 333 K |
Highly stratified reservoirs cause uneven displacement in this simulation. Although CO_{2} is injected in the lower layer (k_{3} = 200 mD), it travels much faster in the upper layer (k_{1}) compared to the other two layers. We can speculate that a serious gas breakthrough will happen in the upper layer (k_{1}). In this case, mobility control measures like water alternating gas or simultaneous water alternating gas will be helpful, and certainly, the adverse effect of CO_{2} solubility in water should be more pronounced than in this example. Moreover, neglecting gravity may fail to consider the upward migration of CO_{2} due to the buoyancy, thereby mispredicting the development of CO_{2} flooding. For example, if CO_{2} is injected at the bottom of the oil-bearing layer where the permeability is relatively high, neglecting the gravity would underestimate the oil recovery because the simulation wrongly predicts a quick gas breakthrough in the bottom layer. On the contrary, such as the run_SPE1, neglecting the gravity would overestimate the performance because the simulation delays the gas breakthrough in the upper layer. For a high permeable reservoir, the gravity effect is trivial because CO_{2} moves much faster horizontally than vertically, but for a unconventional formation, considering the effect of gravity is recommended to achieve an accurate simulation result.
4.3 The efficiency of the formulation
Newton iteration per time step and time ratio in the five runs (time ratio is defined as the ratio of simulation time of the PR–WS model and the conventional model)
Time ratio | Conventional model | PR–WS model | |
---|---|---|---|
run1 | 1.54 | 3.86 | 4.61 |
run2 | 1.55 | 4.07 | 5.00 |
run3 | 1.59 | 4.46 | 5.46 |
run_s | 1.60 | 4.07 | 5.00 |
run_SPE1 | 1.57 | 6.07 | 6.61 |
Average | 1.57 | 4.51 | 5.34 |
5 Conclusions
- 1.
PR EOS and the WS mixing rule with the NRTL model are an accurate approach to predict the phase behavior of the binary CO_{2}/water system. The proposed method with salinity effect correction achieves a good match with the experimental data.
- 2.
Selecting natural variables and full implicit natural variables of the PR–WS model enhances efficiency as well as reliability. For the test example of CO_{2} flooding, this model converges in 4–8 iterations per time step and the total simulation time is 57% more than the conventional model.
- 3.
CO_{2} flooding examples show that a significant amount of injected CO_{2} is dissolved in water and is unavailable for mixing with oil. For example, the run3 displays up to 7% of injected CO_{2} is dissolved in the aqueous phase, which results in a delayed oil recovery of 4%.
- 4.
The gas rate of the PR–WS model in all examples is smaller than the conventional model because a significant portion of injected CO_{2} is dissolved in water, which can be considered as “lost” in the aqueous phase.
- 5.
CO_{2} breakthrough happens in advance of the turning point where the delayed oil starts to be recovered. In field applications, the delayed oil is hard to recover due to serious gas breakthrough.
Notes
Acknowledgements
This work was financially supported by National Natural Science Foundation of China (U1762101) and National Science and Technology Major Projects (2017ZX05069). Special thanks for Changqing Oilfield for providing detailed geology and production data. We would like to express our heartfelt gratitude to Professor Haining Zhao for his constructive suggestion about this paper. Computer Modeling Group are also acknowledged for offering CMG software.
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