Using a Multiple-Mixing-Cell Model to Study Minimum Miscibility

1 key thermodynamic equilibrium tie lines: the initial tie line, the injection tie line, and nc - 3 .... temperature and pressure in each cell, (2) no...
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Ind. Eng. Chem. Res. 2006, 45, 7913-7923

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Using a Multiple-Mixing-Cell Model to Study Minimum Miscibility Pressure Controlled by Thermodynamic Equilibrium Tie Lines Gui-Bing Zhao, Hertanto Adidharma,* Brian Towler, and Maciej Radosz Soft Materials Laboratory, Department of Chemical & Petroleum Engineering, UniVersity of Wyoming, Laramie, Wyoming 82071-3295

In this work, we demonstrate that our simulation approach, referred to as the multiple-mixing-cell model, gives the same results as the analytical method for two-phase gas flooding. For a gas-oil system with nc components, the gas injection process leads to nc + 1 constant-composition zones, and the compositional path of the process, i.e., the path in composition space representing the total composition, is controlled by nc - 1 key thermodynamic equilibrium tie lines: the initial tie line, the injection tie line, and nc - 3 crossover tie lines. In addition, our approach clearly demonstrates that both the gas/oil ratio and fluid mobility do not affect tie-line compositional paths, i.e., compositional paths that lie on the thermodynamic equilibrium tie lines, but they do affect nc - 2 non-tie-line compositional paths. Therefore, neither the gas/oil ratio nor the fluid mobility affects the minimum miscibility pressure (MMP). In the context of the multiple-mixing-cell model, this fact has never been explicitly clarified before. Introduction Miscible gas flooding is a well-known methodology for enhanced oil recovery. One of the key parameters in the design of gas flooding is the minimum miscibility pressure (MMP), which is the lowest pressure required to achieve multiple-contact miscibility between the injected fluid and oil at the reservoir temperature. Common approaches developed for calculating MMP include correlations, the negative flash approach, the multiple-contact single-cell method, a one-dimensional composition simulator, the analytical method, and the multiplemixing-cell model. Among these approaches, the analytical method and the multiple-mixing-cell model are the most appealing because they are robust and explanatory for complex miscibility process mechanisms. The analytical approach to calculating MMP has been well established by Orr and co-workers.1-9 By analytically solving a set of simplified material balance equations of gas displacement processes, they found that the compositional path, which is defined as the path in composition space representing the total composition of the process, consists of nc - 1 key tie lines and nc - 2 non-tie lines.2 The multicontact miscibility of gas injection is controlled by a sequence of nc - 1 key tie lines: the initial tie line, the injection tie line, and nc - 3 crossover tie lines. The MMP is therefore the lowest pressure at which any of the key tie lines is a critical tie line. Johns et al.6 showed that, if any two adjacent tie lines are connected by a shock, the extensions of the tie lines must intersect. If the compositional path contains one or more continuous variations between tie lines (rarefactions), then the tie lines obtained by the intersection method are an approximation to the actual key tie lines.9 The multiple-mixing-cell model, as initially developed by Cook et al.10 and Metcalfe et al.,11 has been used to explore the mechanism of the multiple-contact miscibility process. Pederson et al.12 applied a cell-to-cell model with a total of five cells to simulate the injection of various gases into two different North Sea reservoir oils. They showed that pseudoternary diagrams are insufficient to represent the mixing processes. This finding * To whom correspondence should be addressed. Tel.: 307-7662909. Fax: 307-766-6777. E-mail: [email protected].

was further verified by Zick,13 who proposed for the first time a combined condensing/vaporizing mechanism in the displacement of oil by gases. The multiple-mixing-cell model has been further developed by Jaubert’s research group14-16 in France since 1998. They claimed that previous versions of the multiplemixing-cell model cannot be used to properly calculate MMP because all previous algorithms ignored the presence of crossover tie lines. By comparing the multiple-mixing-cell model with a commercial one-dimensional simulator in calculating MMP, they found the computation speed using a multiplemixing-cell model to be 15-80 times faster than that for the one-dimensional simulator. They also proposed two different criteria for determining MMP. Upon increasing injection pressure, a pressure is defined as the MMP when the oil recovery at 1.2 pore volumes of gas injection first reaches 97% 14 or when the density difference of two phases somewhere in the slim tube first reaches 0.15 The slim-tube experiment is one of the most widely used techniques and is accepted as a standard means for MMP measurement in the petroleum industry. Unlike consolidated core flooding, a slim tube packed with sand or glass beads largely eliminates the effects of adverse dispersion, viscous fingering, capillary pressure, and interfacial tension on multiple-contact miscibility development in gas injection.14,17 Under such conditions, MMP is controlled only by the thermodynamic properties of the injected gas and the original oil in place. Therefore, a one-dimensional multiple-mixing-cell model with multicell equilibrium P/T flash calculations should well represent the process of miscibility development and composition evolution in a slim tube without considering dispersion and interfacial tension. In a multiple-mixing-cell model, several important parameters are the number of cells, the volume ratio of gas added to the first cell to the original oil in the cell (i.e., gas/oil ratio, GOR), and the fluid fractions moved from one cell to the next (phase mobility). Studies on the effects of these parameters on the compositional path in a slim tube would provide deeper insight into the development of miscibility. In the past, about 10 cells were used to simulate the slim tube because of computer memory limitations.10,11 Recently, Jaubert et al.14,15 showed that 500-1000 cells were needed to calculate oil recovery. The only study that investigated the effects of phase mobility and GOR

10.1021/ie0606237 CCC: $33.50 © 2006 American Chemical Society Published on Web 10/07/2006

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Figure 1. Schematic diagram of cell-to-cell simulation.

on the development of miscibility was performed by Metcalfe et al.11 This study showed that both GOR values greater than 0.2 and phase mobility have significant effects on the development of miscibility. Jaubert et al.14,15 predicted the MMP using a GOR value of 0.3 for different gas injections into various oils. Since then, no detailed studies have been reported on how these two parameters affect the compositional path in a slim tube and, thus, the MMP. Therefore, the goals of this work are to investigate the effects of GOR and phase mobility on the compositional path and miscibility development as gas is injected into a slim tube. This will clarify previous works done by Metcalf et al.11 and Jaubert et al.14,15 Multiple-Mixing-Cell Model A multiple-mixing-cell model is a discrete model of a continuous gas injection process in the slim-tube experiment. A packed slim tube is discretized into a series of constantvolume cells (Ci), and the continuous gas injection process is discretized into a series of constant-volume batches (Bi) as shown in Figure 1. With the assumptions of (1) constant temperature and pressure in each cell, (2) no dispersion among cells, (3) no capillary force within each cell or between cells, and (4) perfect mixing in each cell, so that thermodynamic equilibrium can be attained if the fluid in the cell is thermodynamically unstable, a multiple-mixing-cell model is converted into a pure thermodynamic P/T flash calculation. A blockalgebra simultaneous flash algorithm18 coupled with the PengRobinson (PR) cubic equation of state (EOS) is used in this work. However, several parameters should be predetermined before calculation. They are (1) the total number of cells (Nc), (2) the amount of gas added to cell 1 in each batch (GOR), and (3) the fluid fractions moved from cell to cell after the flash calculation. The fluid fractions moved from cell to cell are determined from the fractional flow function for gas (vapor)-oil (liquid) two-phase flow2

(1)

where kr is the relative permeability, µ is the viscosity, and the subscripts g and o refer to gas and oil, respectively. Phase mobility is defined as krgµo/kroµg,11 which is the ratio of the fractional flow function for the two phases in eq 1. The relative permeabilities of the gas and liquid phases used here are6

krg ) kro )

Sg2 Sg2 + (1 - Sg - Sor)2 (1 - Sg - Sor)2 Sg2 + (1 - Sg - Sor)2

fg ) fo )

Sg2/µg Sg2/µg + (1 - Sg)2/µo (1 - Sg)2/µo Sg2/µg + (1 - Sg)2/µo

(2)

where S is the fluid saturation in each cell and the subscript or refers to the residual oil. The magnitude of the residual oil saturation depends primarily on mixing caused by diffusion/

(3)

For constant cell volume, fg falls between 1 and Sg2/[Sg2 + (1 -Sg)2], which corresponds to the limits of µg , µo (i.e., infinite gas-phase mobility) and µg ) µo, respectively. These limits are also consistent with the actual gas flooding process at pressures far lower than the MMP where the difference in properties between gas and oil is huge (µg , µo) and at pressures close to the MMP where the difference in properties between gas and oil somewhere in the slim tube is pretty small (µg ) µo). The upper limit of fg is also identical to the moving-excessoil option presented by Metcalfe et al.11 and used by Jaubert et al.14-16 At this limit, if two phases are present, the gas phase is moved to the next cell first, and if, after all of the gas phase has been moved, the liquid volume is still larger than the cell volume, part of the liquid phase is then moved. Whereas Metcalfe et al.11 used a real phase mobility to determine the fractional flow in their cell-by-cell calculations, we will discuss how the compositional path evolves as the fractional flow function varies from 1 to Sg2/[Sg2 + (1 - Sg)2]. The calculation procedure of the gas displacement process in the slim-tube experiment is as follows: (1) Set the total number of cells, Nc, and an arbitrary cell volume, Vc (cell volume does not affect the end results). All of the cells initially contain the amount of original reservoir oil. (2) Set the GOR. The amount of gas injected into the first cell in each batch is equal to GOR × Vc. As discussed earlier, different investigators have used different numbers of cells and gas/oil ratios for their calculations. The total number of batches of gas injections, Nb, can be calculated from

N b ) Nc

krg/µg fg ) krg/µg + kro/µo kro/µo fo ) ) 1 - fg krg/µg + kro/µo

dispersion and should be very small, less than about 5% of the pore volume in miscible gas reservoir floods and even less in slim-tube displacement, as discussed by Stalkup.19 With the assumptions of no dispersion and perfect mixing, Sor is exactly 0. Therefore, the fractional flow functions in eq 1 become

1.2 GOR

(4)

where the value of 1.2 is a widely accepted criterion, i.e., the required amount of gas injected for oil recovery calculations from slim-tube experiments is 1.2 times the pore volume. (3) Gas in the amount of GOR × Vc is injected into cell 1. Assuming perfect mixing, the vapor fraction in the cell (V) can be determined from a P/T flash calculation. If V g 1, the mixture is at or above its dew point, and the total gas volume must be larger than the cell volume. The excess gas is then moved into cell 2. If V e 0, the mixture is at or below its bubble point. The excess oil is moved into cell 2. If 0 < V < 1, the mixture is in the two-phase region. The excess fluid moved from cell 1 to cell 2 is determined by the fractional flow function, i.e., eq 3, as described earlier. The liquid saturation (So) is defined as

So )

Vl Vl + V v

(5)

where Vl and Vv are the volumes of the liquid and vapor phases, respectively, after the flash calculation. The tie-line length (TL) is defined as

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Figure 2. Compositional path of a ternary system by concentration-space analysis. Table 1. Component Properties for the Three-Component System binary interaction parameters, kij component

Mw

Pc (MPa)

Tc (K)

ω

C1

C4

C10

C1 C4 C10

16.04 58.12 142.29

4.599 3.796 2.110

190.56 425.12 617.70

0.011 0.200 0.490

0 0.0139 0.044

0.0139 0 0.010

0.044 0.010 0

TL )

x∑

(xi - yi)2

(6)

i

where xi and yi are the equilibrium mole fractions of component i in the liquid and vapor phases, respectively. (4) The excess volume formed in cell 1 is transferred to cell 2 and so on until oil is recovered from the last cell. (5) When one batch calculation has been completed, a new injection into cell 1 is performed, and the cell-to-cell calculations, i.e., steps 3 and 4, are repeated. (6) The oil recovery can be calculated from the following equation

RF1.2Nc )

Vr Vo

|

(7) 1 atm,298 K

where Vr is the liquid volume of the fluids recovered from cell Nc measured at 1 atm pressure and 298 K after 1.2 total cell volumes (pore volumes) of injected gas and Vo is the total volume of the original oil in the slim tube at 1 atm pressure and 298 K. Results and Discussion In the following discussion, we use three-, four-, and fivecomponent systems to investigate the effects of the total number of cells, the GOR, and the fractional flow on the compositional path and the process of miscibility development. There are two ways to observe the composition path evolution from a multiplemixing-cell model. One is to analyze the composition profile along the slim tube (cells) at a certain batch number, which

corresponds to a snapshot of the concentration profile at a certain time. We refer to this approach as the concentration-space analysis. The other is to analyze the composition change with time (increasing batch number) in a certain cell. We refer to this approach as the concentration-time analysis. Three-Component System. The example of a ternary system selected in this work is pure methane injection into 50 mol % n-C4H10 and 50 mol % n-C10H22. The component properties and PR EOS binary interaction parameters are reported in Table 1.20 If we divide the slim tube into 1000 cells, set GOR ) 0.3, and set fg ) 1 (moving-excess-oil option with the limit of µg , µo), Figure 2 shows how the composition paths evolve for different numbers of batches at 344 K and 16 MPa. All composition paths for different batches are strictly bounded between the initial and injection tie lines. At batch 2, the injected gas is still completely dissolved in the oil, and therefore, the compositional path does not yet cross the two-phase region. As also shown in the figure, the light species, CH4, is richer in the oil at the upstream side than at the downstream side of the slim tube. As the amount of gas injected increases, at batch 5, the fluid splits into two phases in the first cell, but not in the succeeding cells. As shown in the figure, at batch 5, there is only one point in the two-phase region. As the amount of gas injected is increased further, increasing numbers of cells in the upstream side of the slim tube contain two phases, and therefore, increasing numbers of points of the compositional path are in the two-phase region. When the number of batches is increased to 50, the fluid in the first cell changes from ternary to a binary system. This is because the equilibrium constant (Ki, the ratio of the mole fraction of component i in the vapor phase to that in the liquid phase, yi/xi) of C4 is larger than that of C10, which

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Figure 3. Compositional path of a ternary system by concentration-time analysis.

Figure 4. Comparison of compositional paths of a ternary system obtained from two different approaches at steady state: composition-space and compositiontime.

causes C4 to flow downstream faster than C10 and to separate from C10. When the number of batches is increased to 200, C10 in the first cell disappears, and the fluid in the first cell becomes pure methane, which is the injected gas. When the number of batches is greater than 200, the composition path does not change and thus reaches a steady state. In the composition space such as in Figure 2, from upstream to downstream of the slim tube, the compositional path first moves from point A representing the composition of the injected gas to point B in the twophase region along the injection tie line, up to which only binary systems exist. Then, the compositional path moves toward point C, which lies on the initial tie line, and finally reaches point D, representing the composition of the original oil. Figure 3 shows how the compositional path evolves with increasing numbers of batches at different cells. All of the compositional paths at different cells are bounded by the initial

and injection tie lines. For the first several batches, in the first cell, methane is dissolved into the oil, and thus, only a liquid phase exists. As the amount of gas injected (i.e., the number of batches) increases, the fluid in the first cell changes from a liquid phase to vapor and liquid phases and then finally to a gas phase. In the liquid-phase region, the compositional path evolves along the line connecting the compositions of the initial oil and the injected gas. In the successive cells, the compositional paths of the liquid phase finally evolve along the initial tie line. After the fifth cell, the compositional paths of all cells are exactly identical, which means that the steady state has been reached. Figure 4 shows a comparison of the compositional paths obtained from the composition-space (for 600 batches) and composition-time (at cell 20) analyses at steady state. The composition paths obtained from the two approaches are exactly the same. Therefore, we can choose to observe composition

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Figure 5. Results of cell-to-cell simulation at batch 2000 for pure methane injection into 50% C4 + 50% C10 with GOR ) 0.3 and fg ) 1 at 344 K and 16 MPa. (a) Composition profile along the slim tube: s, C1; - -, C4; - ‚ -, C10. (b) Density and liquid saturation profiles along the slim tube: s, liquid saturation; - -, liquid density; - ‚ -, vapor density. (c) Equilibrium ratio and tie-line length profiles along the slim tube: s, C1; - -, C4; - ‚ -, C10; - - -, tie-line length.

Figure 6. Compositional paths for different gas/oil ratios at fg ) 1 (symbols, constant-composition zones; dashed lines, compositional paths). GOR ) 0, 0.0001; O, 0.001; 4, 0.3; 3, 0.6; ], 1.5; right-pointing triangles, 3; left-pointing triangles, 10; g, 200.

evolution using either composition-space or composition-time analysis, which should give the same results on the miscibility development of the gas displacement process in the slim tube. For the rest of this work, we discuss the calculation results on composition evolution using a composition-space analysis. As shown in Figure 4, note that the compositional path of the gas displacement process includes the two tie-line paths AB and CD and the non-tie-line path BC; a tie-line path is defined as the compositional path along a tie line. Figure 5a shows the composition profile in the slim tube at batch 2000. It is clear that the slim tube is divided into four zones of constant composition, with short transition zones between adjacent zones of constant composition where the composition of the fluid changes from cell to cell. In Figure 4, these four zones are represented by four points: A, B, C, and D. The fluids in zone 1 at the upstream side of the slim tube have the

same composition as the injected gas (i.e., pure methane). The fluids in zones 2 and 3 are binary (i.e., methane + decane) and ternary mixtures, respectively. Finally, in zone 4, the fluids have the same composition as the original oil. The density (F) and liquid saturation (So) profiles in Figure 5b show a similar trend. In zone 1, So and the liquid density are 0 because only gas is present. In zone 4, So is 1, and the vapor density is 0 because only liquid phase is present. In each zone, the fluid density and liquid saturation are very different. Figure 5c shows the profile of the equilibrium ratio Ki and the tie-line length TL along the slim tube. Because zones 1 and 2 lie on the same injection tie line and zones 3 and 4 lie on the same initial tie line, as shown in Figure 2, there are only two zones of constant value for Ki and TL. Figure 6 shows the effect of the GOR on the compositional path for an fg value of 1. It is clear that the GOR does not affect

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Figure 7. Oil recovery as a function of total number of cells at different gas oil ratios.

Figure 8. Comparison of the compositional paths at two limits of the fractional flow function. Constant-composition zones: 0, µg , µo; O, µg ) µo. Compositional paths: - -, µg , µo; ‚ ‚ ‚, µg ) µo.

the tie-line path, but it does affect the non-tie-line path (BC in Figure 4). The mole fractions of all constant-composition zones at different GORs lie on the key tie lines, i.e., the initial tie line and the injection tie line. For high GORs, the non-tie-line path is close to the dew-point curve. For low GORs, the non-tieline path is close to the bubble-point curve. The non-tie-line path moves along the dew-point curve at the limit of GOR f ∞, whereas the non-tie-line path moves along the bubble-point curve at the limit of GOR f 0. This is due to the high vapor fraction in the two-phase region for high GORs (more gas is injected in each batch). Figure 7 shows the effects of the total number of cells and GOR on the oil recovery for an fg value of 1. It is clear that more cells are required to obtain constant oil recovery, i.e., oil recovery that does not depend on the total number of cells, with increasing GOR. For example, 6000 cells are required to obtain constant oil recovery for GOR ) 100, 3000 cells are required for GOR ) 10, and 500 cells are required for GOR ) 0.05. As shown in Figure 6, the larger constant oil recovery is for the lower GOR. The reason for the larger constant oil recovery at the lower GOR is that the non-tie line path is closer to the

bubble-point curve, where the vapor fraction is smaller and the liquid saturation is higher. Figure 8 compares the compositional paths at two limits of the fractional flow function, i.e., 1 and Sg2/[Sg2 + (1 - Sg)2], which correspond to µg , µo and µg ) µo, respectively. The compositional path for any realistic fractional flow function should fall between the compositional paths of these two limits. Obviously, fractional flow functions do not affect the tie-line path, but they do affect the non-tie-line path (BC and B′C′). The mole fractions of all constant-composition zones (points A, B, B′, C, C′, and D) lie on the key tie lines. For the same GOR, the non-tie-line path is closer to the dew-point curve at the limit of µg ) µo (the mole fractions of the lightest species CH4 at points B′ and C′ are higher than those at points B and C, respectively), because liquid flow is accelerated and gas flow is slowed at the limit of µg ) µo, which causes a larger volume of vapor phase (including more of the lightest species CH4) in the cell and makes the vapor fraction at the non-tie-line path higher than that for the limit of µg , µo. The area of B′BCC′ becomes smaller with increasing pressure, and points C and C′

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eventually coincide at the MMP where the initial tie line is tangent to the critical point. There are many criteria that can be used to determine the gas injection pressure at which miscibility has been achieved. Four well-known criteria have been presented: (1) whether the oil recovery is higher than 97%,14 (2) whether the gas-liquid density difference is close to 0,15 (3) whether the compositional path goes through the critical point of fluid in the slim tube,11 and (4) whether any key tie-line length is close to 0.8 As shown in Figure 7, oil recovery is not a good criterion for miscibility because it depends on the GOR. The gas-liquid density difference should be used with care because there are singlephase regions along the slim tube where no density difference exists. Although compositional path could theoretically be a good criterion for miscibility, it is not a direct approach to judge whether a fluid is at the critical point, especially for multicomponent systems. A direct way to determine the miscibility of fluids is to check the tie-line length. Even in the single-phase region at pressures below the critical pressure, the tie-line length can be determined from negative flash. If the tie-line length of a fluid is 0, the fluid is at the critical point. As shown in Figures 6 and 8, both the GOR and the fractional flow function affect only the non-tie-line path, but not tie-line paths. Therefore, tieline length is a robust criterion for determining the miscibility of the gas injection process. For a ternary system (nc ) 3), it is clear that there are four (nc + 1) constant-composition zones and that the compositional path is controlled by two (nc - 1) key tie lines. The compositional path consists of two (nc - 1) key tie-line paths and one (nc - 2) non-tie-line path. Before we study more complex systems, it is also worth mentioning that the GOR and the fractional flow function apparently induce numerical dispersion in our model. From the analytical solution,2,8 for a self-sharpening system, in the absence of physical dispersion, all key tie lines are connected by shocks in the composition profile. Because our ternary system is a typical self-sharpening system, the transition between key tie lines should, in fact, be represented by a shock in the composition profile. As shown in Figure 5, the transition obtained from our model is not strictly a shock, but a smeared transition zone. This indicates that our approach does induce numerical dispersion. Jessen et al.21 investigated the evolution of the compositional path for a similar ternary system using numerical finite-difference schemes. They found that a small number of gridblocks caused so large a numerical dispersion that the key tie lines could not be identified. For our multiplemixing-cell model, the cell volume does not affect the compositional path as indicated before. The two factors in our model that induce numerical dispersion are the GOR and the fractional flow function. Figure 9 shows the extent of numerical dispersion induced by the GOR with the same amount of gas injected (number of batches × GOR). It is clear that the length of the transition zone (non-tie-line path) between two key tie lines increases with increasing GOR. We also observe that the transition zone between zones 3 and 4 is smeared as well. Therefore, a large GOR would cause a large numerical dispersion. Although a low GOR produces a transition zone close to a shock, a low GOR requires a large number of batches, which increases the computation time. The acceptable value of the GOR is in the range of 0.1-1 depending on the phase behavior of the system. Figure 10 compares the extent of numerical dispersion induced by the two limits of the fractional flow function. The numerical dispersion for the fractional flow function at the limit of µg ) µo is slightly larger than that at the

Figure 9. Extent of numerical dispersion induced by the GOR for the same amount of gas injected (number of batches × GOR) for pure methane (1) injection into 50% C4 (2) + 50% C10 (3) for fg ) 1 at 344 K and 16 MPa. s, GOR ) 0.05 and batch 12000; - -, GOR ) 0.3 and batch 2000; ‚ ‚ ‚, GOR ) 1.5 and batch 400; - ‚ -, GOR ) 10 and batch 60; - ‚ ‚ -, GOR ) 100 and batch 6.

Figure 10. Extent of numerical dispersion induced by the two limits of the fractional flow function for pure methane (1) injection into 50% C4 (2) + 50% C10 (3) for GOR ) 0.3 at 344 K and 16 MPa. Compositional paths: s, µg , µo; - -, µg ) µo.

limit of µg , µo. When the GOR or fractional flow function is changed, the mole fractions at zones of constant composition are also changed, as also shown in Figures 6 and 8. Although numerical dispersion exists in our multiple-mixing-cell model, it does not affect the identification of key tie lines, provided that a steady state has been reached. Four-Component System. Another example selected in this work is a quaternary system, in which a gas mixture of 61.5% CH4 + 38.5% n-C3 is injected into a slim tube containing oil with a composition of 20% CH4 + 40% n-C6 + 40% n-C16. This system has been well investigated by Johns et al.6 The component properties and PR EOS binary interaction parameters are listed in ref 6. As for ternary systems, after steady state is reached, the compositional paths for both the composition-space and composition-time analyses are exactly the same. Figure 11

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Figure 11. Results of cell-to-cell simulation at batch 1600 for 61.5% C1 + 38.5% C3 injected into 20% C1 + 40% C6 + 40% C16 at 366.5 K and 12 MPa. (a) Composition profile along the slim tube: s, C1; - -, C3; ‚ ‚ ‚, C6; - ‚ -, C16. (b) Density and liquid saturation profiles along the slim tube: s, liquid saturation; - -, liquid density; - ‚ -, vapor density. (c) Equilibrium ratio and tie-line length profiles along the slim tube: s, C1; - -, C3; ‚ ‚ ‚, C6; - ‚ -, C16; - ‚ ‚ -, tie-line length.

Figure 12. Compositional paths at different gas/oil ratios for 61.5% C1 + 38.5% C3 injected into 20% C1 + 40% C6 + 40% C16 at 366.5 K and 12 MPa (symbols, constant-composition zones). GOR: 0, 0.02; O, 0.3; 4, 1.5; 3, 10; ], 30.

shows the results of our simulation using the multiple-mixingcell model at batch 1600 with GOR ) 0.3 and fg ) 1 at 366.5 K and 12 MPa. It is clear that the slim tube is divided into five zones (nc + 1) of constant composition, i.e., zones containing injected gas, C1 + C3 + C16, C1 + C3 + C6 + C16, C1 + C6 + C16, and original oil, as shown in Figure 11a. At pressures below the MMP, the whole process of gas injection is pretty similar to a gas stripping process, i.e., the light species of the original oil move downstream and leave the heavy species at the upstream side of the slim tube. Within each constant-composition zone, the fluid density and liquid saturation are constant, as shown in Figure 11b. The K value of each species and the tie-line length are shown in Figure 11c. Because zones 1 and 2 share the same injection tie line and zones 4 and 5 share the same initial tie line, there are three zones of constant K value and tie-line length in the slim tube. Figure 12 shows the compositional paths with fg ) 1 at different GOR values. It is clear that all of the compositional

Figure 13. Effects of fractional flow on the compositional path for 61.5% C1 + 38.5% C3 injected into 20% C1 + 40% C6 + 40% C16 at 366.5 K and 12 MPa. 0, µg , µo; O, µg ) µo.

paths are controlled by three (nc - 1) key tie lines, i.e., the initial tie line, the injection tie line, and the crossover tie line, as defined by Monroe et al.4 The composition first moves along the injection tie line from that of the injected gas (zone 1 in Figure 11a) to a two-phase region in the C1C3C16 plane until constant-composition zone 2, then moves to the crossover tie line until constant-composition zone 3 (this is the non-tie line path), then moves to the initial tie line until constant-composition zone 4 (this is the non-tie line path), and finally moves along the initial tie line to the composition of the original oil (zone 5). Therefore, the compositional path of a quaternary system consists of three key tie-line paths and two non-tie-line paths (nc - 2). Moreover, as in the ternary system, the GOR affects non-tie-line paths, but not tie-line paths. Also, the non-tie-line path is close to the dew-point curve for high GOR and to the bubble-point curve for low GOR. The non-tie-line path moves along the dew-point curve at the limit of GOR f ∞, and it

Ind. Eng. Chem. Res., Vol. 45, No. 23, 2006 7921 Table 2. Component Properties for the Five-Component System binary interaction parameters, kij component

Mw

Pc (MPa)

Tc (K)

ω

C1

C3

C4

C6

C10

C1 C3 C4 C6 C10

16.04 44.09 58.12 86.18 142.29

4.599 4.246 3.796 2.969 2.110

190.56 369.73 425.12 507.39 617.70

0.011 0.153 0.200 0.299 0.490

0 0 0.027 0.025 0.042

0 0 0 0.010 0

0.027 0 0 0 0.008

0.025 0.010 0 0 0

0.042 0 0.008 0 0

Table 3. Compositional Paths of the Five-Component System at Different GOR Values and at Two Limits of the Fractional Flow Function of 1 and Sg2/[Sg2 + (1 - Sg)2] for GOR ) 1.5 Tie Line I (Injection Tie Line) GOR component C1 C3 C4 C6 C10

z at zone I (gas)

x

0.9 0.1 0 0 0

0.301 0.164 0 0 0.535

0.3 0.606 0.132 0 0 0.259

1.5

1.5a z at zone II

0.729 0.118 0 0 0.153

0.835 0.107 0 0 0.0581

10

100 y

0.851 0.105 0 0 0.0435

0.888 0.101 0 0 0.0111

0.892 0.101 0 0 0.00631

Tie Line II (Crossover Tie Line) GOR 0.3 component

1.5a

1.5

x

C1 C3 C4 C6 C10

0.311 0.163 0 0.191 0.336

10

100

z at zone III 0.514 0.141 0 0.129 0.216

0.654 0.126 0 0.0870 0.133

0.784 0.112 0 0.0474 0.0555

y 0.813 0.109 0 0.0390 0.0389

0.864 0.104 0 0.0229 0.00891

0.871 0.103 0 0.0216 0.00483

Tie Line III (Crossover Tie Line) GOR component

1.5a

0.3

1.5

0.378 0.167 0.202 0.0896 0.164

0.516 0.150 0.163 0.0635 0.108

x

C1 C3 C4 C6 C10

0.315 0.174 0.220 0.101 0.189

10

100

z at zone IV 0.598 0.137 0.135 0.0456 0.0693

y 0.698 0.128 0.111 0.0294 0.0345

0.765 0.118 0.0929 0.0176 0.00870

0.771 0.119 0.090 0.0158 0.00508

Tie Line IV (Initial Tie Line) GOR 0.3 component

x

C1 C3 C4 C6 C10

0.325 0 0.290 0.132 0.253

0.390 0 0.268 0.118 0.223

1.5

1.5a z at zone V

0.535 0 0.219 0.0873 0.158

0.618 0 0.191 0.0698 0.120

10

100 y

0.722 0 0.156 0.0479 0.0731

0.788 0 0.131 0.0327 0.0405

0.875 0 0.105 0.0159 0.00427

z at zone VI (oil) 0 0 0.4 0.2 0.4

a Results at the limit of the fractional flow function of S 2/[S 2 + (1 - S )2] corresponding to µ ) µ ; other results were obtained from the limit of the g g g g o fractional flow function of 1 corresponding to µg , µo.

moves along the bubble-point curve at the limit of GOR f 0. The effects of the total number of cells and the GOR on oil recovery for the quaternary system are the same as for the ternary system. With a fixed GOR of 0.3, Figure 13 shows the compositional paths at two limits of the fractional flow function, i.e., 1 and Sg2/[Sg2 + (1 - Sg)2]. The compositional path for any realistic fractional flow function should fall between the compositional paths of these two limits. Again, the fractional flow function does not affect the tie-line paths, but it does affect the non-tieline paths. As the injection gas pressure is increased, the miscibility of multiple-contact gas injection is achieved when the tie-line length of one of the three key tie lines reaches 0. The same system has been investigated by Johns et al.6 using an analytical method. They argued that the crossover tie line intersected both the injection and initial tie lines. Linear

equations of the three key tie lines shown in Figures 12 and 13 are known from our multiple-mixing-cell model. The injection tie line is the line connecting point xi ) (0.304, 0.508, 0, 0.188) and point yi ) (0.611, 0.386, 0, 0.00225); the crossover tie line is the line connecting point xi ) (0. 343, 0.503, 0.0903, 0.0637) and point yi ) (0.545, 0.417, 0.0348, 0.00330); the initial tie line is the line connecting point xi ) (0.382, 0, 0.313, 0.305) and point yi ) (0.967, 0.0323, 0, 0.000283). Our results confirm that the crossover tie line intersects both the injection and initial tie lines. The intersection point of the injection tie line and the crossover tie line is the point (0.672, 0.362, 0, -0.0346), and the intersection point of the crossover tie line and the initial tie line is the point (1.52, 0, -0.233, -0.288). Five-Component System. Another example selected in this work is a five-component system, in which a gas mixture of 90% CH4 + 10% n-C3 is injected into a slim tube containing

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Ind. Eng. Chem. Res., Vol. 45, No. 23, 2006

Figure 14. Results of cell-to-cell simulation at batch 5000 for a fivecomponent system for GOR ) 0.3 and fg ) 1 at 366.5 K and 10 MPa. (a) Composition profile along the slim tube. (b) Equilibrium liquid composition profile along slim tube. (c) Equilibrium vapor composition profile along the slim tube: s, C1; - -, C3; ‚ ‚ ‚, C4; - ‚ -, C6; - ‚ ‚ -, C10.

oil of composition 40% n-C4 + 20% n-C6 + 40% n-C10. The component properties and PR EOS binary interaction parameters are listed in Table 2. Figure 14 shows the evolution of the compositional path of the five-component system after steady state has been achieved. The compositional path consists of six (nc + 1) constantcomposition zones. This compositional path is controlled by four (nc - 1) key tie lines. Zones 1 and 2 share the same tie line I (injection tie line), and zones 5 and 6 share the same tie line IV (initial tie line). Therefore, there are three (nc - 2) nontie line paths, connecting zones 2 and 3, 3 and 4, and 4 and 5.

Table 3 reports the effects of the GOR and the fractional flow function on the compositional path. Again, variation of the GOR from 0.3 to 100 does not change the tie-line paths (all compositions at each constant zone lie on the key tie lines), but it does change the non-tie-line paths (the compositions in each constant zone are different for different GORs). As for other systems, the fractional flow function does not change the tie-line paths, but it does change the non-tie-line paths. For an fg value of 1, as the GOR is increased, the compositions in the constant-composition zone move from the liquid-phase composition to the vapor-phase composition, which demonstrates the earlier results that the non-tie line path is close to the dewpoint curve for high GOR and to the bubble curve for low GOR. At a GOR value of 1.5, the composition in each constantcomposition zone moves from the liquid-phase composition to the vapor-phase composition when the fractional flow function is varied from 1 to Sg2/[Sg2 + (1 - Sg)2]. The intersection of any two adjacent key tie lines is also verified for this five-component system. For example, the intersection point of key tie lines I (injection tie line) and II is the point (0.968, 0.0927, 0, 0, -0.0612), the intersection point of key tie lines II and III is the point (1.09, 0.0802, 0, -0.0434, -0.122), and the intersection point of key tie lines III and IV (initial tie line) is the point (1.75, 0, -0.188, -0.167, -0.389). Non-Self-Sharpening System. All of the previous examples are self-sharpening systems, for which, from the analytical solution, any two adjacent key tie lines are connected by a shock. Our multiple-mixing-cell model can accurately identify all key tie lines for those systems. To support the generality of our conclusion, by using our multiple-mixing-cell model, we also investigated a typical non-self-sharpening system, i.e., 50% N2 + 50% CH4 injected into 50% CH4 + 16.24% C4 + 33.76% C10. This system was also investigated by Jessen et al.21 Figure 15 shows the results at batch 1000 for GOR ) 0.3 and fg ) 1 at 344.26 K and 25 MPa. The results are similar to those of the quaternary self-sharpening system shown in Figure 11. There

Figure 15. Results of cell-to-cell simulation at batch 1000 for a quaternary non-self-sharpening system for GOR ) 0.3 and fg ) 1 at 344.26 K and 25 MPa. (a) Composition profile along the slim tube. (b) Equilibrium liquid composition profile along the slim tube. (c) Equilibrium vapor composition profile along the slim tube: s, C1; - -, C4; ‚ ‚ ‚, C10; - ‚ -, N2.

Ind. Eng. Chem. Res., Vol. 45, No. 23, 2006 7923

are five zones of constant composition and three key tie lines. The only difference is that the number of cells where the transition zone between any two adjacent key tie lines takes place is larger, which is a typical characteristic for non-selfsharpening systems, i.e., any two adjacent key tie lines are connected by a rarefaction (continuous variation). From Figure 15, the injection tie line is the line connecting points xi ) (0.300, 0, 0.503, 0.197) and yi ) (0.497, 0, 0.00680, 0.496), the crossover tie line is the line connecting points xi ) (0.304, 0.157, 0.334, 0.204) and yi ) (0.479, 0.0455, 0.00803, 0.468), and the initial tie line is the line connecting points xi ) (0.662, 0.122, 0.215, 0) and yi ) (0.917, 0.0596, 0.0231, 0). Our results also demonstrate that the crossover tie line intersects both the injection and initial tie lines even for this non-self-sharpening system. The intersection point of the injection and crossover tie lines is the point (0.550, 0, -0.125, 0.575), and the intersection point of the crossover and initial tie lines is the point (0.169, 0.244, 0.587, 0). From the above results, it is clear that nc - 1 key tie lines control the miscibility development of the gas flooding process for an nc-component system. These key tie lines are the injection tie line, the initial tie line, and nc - 3 crossover tie lines, and any two adjacent key tie lines intersect. The MMP is reached when the tie-line length of any one of nc - 1 key tie lines reaches 0. The compositional path consists of nc - 1 tie-line paths and nc - 2 non-tie-line paths. These results are in agreement with those obtained by Orr’s research group.1-9 Moreover, the key tie lines are GOR- and fractional-flowfunction-independent. This indicates that the key tie lines can be robustly found through our multiple-cell-mixing model. For each key tie line, there is a constant-composition zone. There are two constant-composition zones, i.e., zones having compositions of the injected gas and original oil, that lie on the extensions of the injection and initial tie lines. Therefore, the total number of constant-composition zones is nc + 1 for an nc-component system. Thus, the number of constant-composition zones increases with increasing number of components, which requires more cells to obtain a steady state. The number of cells needed depends, of course, on the specific phase behavior of the system. In this work, 100, 250, and 400 cells are required to show the whole compositional path for three-, four-, and fivecomponent systems, respectively. For more than 10 components, 1000 cells might not be enough to show all of the constantcomposition zones and key tie lines, as indicated by Jaubert et al.15 However, the number of cells required for the non-tie line paths is small, usually less than 100 cells, as shown in Figures 5, 11, and 14. In addition, as the number of batches is increased, the zones and key tie lines appear in reverse order, i.e., the initial tie line appears first, and the injection tie line appears last. This provides a very good hint to quickly find the key tie lines. Once one key tie line (one constant-composition zone) appears, one does not need to calculate the downstream displacement after this constant-composition zone. In this way, one can easily find a new key tie line, and the calculation proceeds quickly. Further work is underway to build a new algorithm to predict the MMP based on our multiple-mixing-cell model. Conclusions For three-, four-, and five-component systems, our multiplemixing-cell model shows the same results for gas flooding processes as the analytical approach. For an nc-component system, the compositional path consists of nc - 1 key tie lines and nc - 2 non-tie-line paths. The minimum miscibility pressure is the pressure at which the tie-line length of any of the key tie

lines reaches 0 as the pressure is increased. Because neither the gas/oil-ratio nor the fractional flow function affects the key tie-line path, neither of them affects the MMP. Our multiplemixing-cell model provides a robust method to determine all key tie lines of gas flooding processes. Literature Cited (1) Orr, F. M., Jr.; Pande, K. K. Effect of multicomponent, multiphase equilibria in gas injection processes. Fluid Phase Equilib. 1989, 52, 247261. (2) Orr, F. M., Jr.; Dindoruk, B.; Johns, R. T. Theory of Multicomponent Gas/Oil Displacements. Ind. Eng. Chem. Res. 1995, 34 (8), 2661-2669. (3) Orr, F. M., Jr.; Johns, R. T.; Dindoruk, B. Development of miscibility in four-component carbon dioxide floods. SPE ReserVoir Eng. 1993, 8 (2), 135-142. (4) Monroe, W. W.; Silva, M. K.; Larsen, L. L.; Orr, F. M., Jr. Composition paths in four-component systems: Effect of dissolved methane on 1D CO2 flood performance. SPE ReserVoir Eng. 1990, 5 (3), 423-432. (5) Johns, R. T.; Orr, F. M., Jr. Miscible Gas Displacement of Multicomponent Oils. SPE J. 1996, 1, 39-50. (6) Johns, R. T.; Dindoruk, B.; Orr, F. M., Jr. Analytical Theory of Combined Condensing/Vaporizing Gas Drives. SPE AdV. Technol. Ser. 1993, 1 (2), 7-16. (7) Dindoruk, B.; Orr, F. M., Jr.; Johns, R. T. Theory of Multicontact Miscible Displacement with Nitrogen. SPE J. 1997, 2 (3), 268-279. (8) Wang, Y.; Orr, F. M., Jr. Analytical calculation of minimum miscibility pressure. Fluid Phase Equilib. 1997, 139 (1-2), 101-124. (9) Wang, Y.; Orr, F. M., Jr. Calculation of minimum miscibility pressure. J. Pet. Sci. Eng. 2000, 27 (3-4), 151-164. (10) Cook, A. B.; Walker, C. J.; Spencer, G. B. Realistic K values of C7+ hydrocarbons for calculating oil vaporization during gas cycling at high pressures. J. Pet. Technol. 1969, 21, 901-915. (11) Metcalfe, R. S.; Fussell, D. D.; Shelton, J. L. A multicell equilibrium separation model for the study of multiple contact miscibility in rich-gas drives. SPE J. 1973, 13 (3), 147-155. (12) Pedersen, K. S.; Fjellerup, J.; Thomassen, P. Studies of Gas Injection Into Oil Reservoirs by a Cell-to-Cell Simulation Model. SPE J. 1986, 15599, 1-8. (13) Zick, A. A. A Combined Condensing/Vaporizing Mechanism in the Displacement of Oil by Enriched Gases. SPE J. 1986, 15493, 1-11. (14) Jaubert, J.-N.; Wolff, L.; Neau, E.; Avaullee, L. A Very Simple Multiple Mixing Cell Calculation To Compute the Minimum Miscibility Pressure Whatever the Displacement Mechanism. Ind. Eng. Chem. Res. 1998, 37 (12), 4854-4859. (15) Jaubert, J.-N.; Arras, L.; Neau, E.; Avaullee, L. Properly Defining the Classical Vaporizing and Condensing Mechanisms When a Gas Is Injected into a Crude Oil. Ind. Eng. Chem. Res. 1998, 37 (12), 48604869. (16) Jaubert, J.-N.; Avaullee, L.; Pierre, C. Is It Still Necessary to Measure the Minimum Miscibility Pressure? Ind. Eng. Chem. Res. 2002, 41 (2), 303-310. (17) Perkins, T. K.; Johnston, O. C.; Hoffman, R. N. Mechanics of Viscous Fingering in Miscible Systems. SPE J. 1965, 5, 301-317. (18) Chen, C.; Duran, M. A.; Radosz, M. Phase equilibria in polymer solutions. Block-algebra, simultaneous flash algorithm coupled with SAFT equation of state, applied to single-stage supercritical antisolvent fractionation of polyethylene. Ind. Eng. Chem. Res. 1993, 32 (12), 3123-3127. (19) Stalkup, F. I. Displacement Behavior of the Condensing/Vaporizing Gas Drive Process. SPE J. 1987, 16715, 1-12. (20) Benmekki, E. H.; Mansoori, G. A. Minimum miscibility pressure prediction with equations of state. SPE ReserVoir Eng. 1988, 3 (2), 559564. (21) Jessen, K.; Stenby, E. H.; Orr, F. M., Jr. Interplay of Phase Behavior and Numerical Dispersion in Finite-Difference Compositional Simulation. SPE J. 2004, 9 (2), 193-201.

ReceiVed for reView May 18, 2006 ReVised manuscript receiVed August 23, 2006 Accepted September 6, 2006 IE0606237