Multiple-Mixing-Cell Method for Three-Hydrocarbon

J169150 DOI: 10.2118/169150-PA Date: 21-July-15

Stage:

Page: 1

Total Pages: 11

Multiple-Mixing-Cell Method for Three-Hydrocarbon-Phase Displacements Liwei Li, Saeid Khorsandi, and Russell T. Johns, Pennsylvania State University; and Kaveh Ahmadi, BP America

Summary Low-temperature oil displacements by carbon dioxide involve complex phase behavior, in which three hydrocarbon phases can coexist. Reliable design of miscible gasflooding requires knowledge of the minimum miscibility pressure (MMP), which is the pressure required for 100% recovery in the absence of dispersion or as defined by slimtube experiments as the “knee” in the recovery curve with pressure in which displacement efficiency is greater than 90%. There are currently no analytical methods to estimate the MMP for multicomponent mixtures exhibiting three hydrocarbon phases. Also, the use of compositional simulators to estimate MMP is not always reliable. These challenges include robustness issues of three-phase equilibrium calculations, inaccurate three-phase relative permeability models, and phase identification and labeling problems that can cause significant discontinuities and failures in the simulation results. How miscibility is developed, or not developed, for a three-phase displacement is not well-known. We developed a new three-phase multiple-mixing-cell method that gives a relatively easy and robust way to determine the pressure for miscibility or, more importantly, the pressure for highdisplacement efficiency. The procedure that moves fluid from cell to cell is robust because it is independent of phase labeling (i.e., vapor or liquid), has a robust way to provide good initial guesses for three-phase flash calculations, and is also not dependent on three-phase relative permeability (fractional flow). These three aspects give the mixing-cell approach significant advantages over the use of compositional simulation to estimate MMP or to understand miscibility development. One can integrate the approach with previously developed two-phase multiple-mixing-cell models because it uses the tie-line lengths from the boundaries of tie triangles to recognize when the MMP or pressure for high-displacement efficiency is obtained. Application of the mixing-cell algorithm shows that, unlike most two-phase displacements, the dispersion-free MMP may not exist for three-phase displacements, but rather a pressure is reached in which the dispersionfree displacement efficiency is maximized. The authors believe that this is the first paper to examine a multiple-mixing-cell model in which two- and three-hydrocarbon phases occur and to calculate the MMP and/or pressure required for high displacement efficiency for such systems. Introduction Slimtube measurements show that oil displacement by carbon dioxide (CO2) involving three hydrocarbon phases can achieve greater than 90% displacement efficiency at temperatures typically less than 120  F (Yellig and Metcalfe 1980; Gardner et al. 1981; Orr et al. 1983). The 1D displacement simulations by Li and Nghiem (1986) showed high oil recovery without a miscible bank. Simulation results of west Texas oil displacement by CO2 (Khan et al. 1992) showed high-displacement efficiency of more than 90% in the presence of immiscible three-hydrocarbon-phase flow. Okuno et al. (2011) explained the mechanism for high-displacement efficiency as the result of the composition path C 2015 Society of Petroleum Engineers Copyright V

This paper (SPE 169150) was accepted for presentation at the SPE Improved Oil Recovery Symposium, Tulsa, 12–16 April 2014, and revised for publication. Original manuscript received for review 31 March 2014. Revised manuscript received for review 17 February 2015. Paper peer approved 30 April 2015.

approaching the critical endpoints (CEPs), as illustrated in Fig. 1. Okuno and Xu (2014a, b) examined further the development of multicontact miscibility in compositional simulation by introducing new distance parameters on the basis of interphase mass transfer near CEPs. CEPs are states in which two of the three coexisting phases merge to a critical point and become identical. There are generally two types of CEPs: The first CEP is where the liquid (L1 and L2) phases merge in the presence of the vapor (V) phase, and the other is where the second-liquid (L2) and V phases merge in the presence of the L1 phase. A CEP is not a point, as it would seem, but is rather a tie line in composition space in which the three-phase region (tie triangle) becomes a two-phase tie-line as one phase vanishes. A tri-critical point is where three phases simultaneously become identical (Widom 1973; Griffiths 1974). This is a true critical point. Oil displacements by CO2 involving L1-L2-V equilibrium can achieve greater than 95% displacement efficiency even if the L1 and V phases by themselves would be significantly immiscible. As explained by Okuno et al. (2011), high displacement efficiency is possible because the L2 phase serves as a buffer between the L1 and V phases. Three-phase compositional simulations typically use only approximate relative three-phase permeability models that commonly do not fit the experimental data well (Delshad and Pope 1989). Guler et al. (2001) performed three-phase simulations and showed that relative-permeability curves effect oil production time but not ultimate recoveries. One task in compositional simulation to model three hydrocarbon phases is to define the threshold phase density to identify and label the phases because the relative permeability models depend on the phase type (Perschke et al. 1989). Phase mislabeling could occur when the unique threshold density fails to label the phases correctly, which can cause discontinuities in the simulation results and subsequent failure (Okuno et al. 2010b). A trial-and-error technique is often applied to identify the best threshold density between the L2 and V phases at relatively high pressure, but this does not always fix the problem. Phase labeling is also important for the reliability of simulation results because it affects the relative permeability model. It is well-known that relative permeability curves should be a continuous function of composition (Jerauld 1997; Blunt 2000). Yuan and Pope (2012) used Gibbs energy to incorporate the effect of phase compositions on relative permeability and showed that doing so eliminated discontinuities in the two-phase simulation displacements near a critical point. Juanes and Patzek (2004) studied the consistency of relative permeability models and defined conditions to make the relative permeability model more physical. Two-phase approximations of three-phase flow are usually used in commercial simulators that do not allow for three hydrocarbon phases; however, this can cause errors in recoveries (Wang and Strycker 2000) and can increase simulation discontinuities and instabilities (Okuno et al. 2010b; Lins et al. 2011). One can avoid many of these problems if the algorithm used in simulation is independent of phase type, but no current simulator was formulated this way. Three-phase equilibrium calculations are typically computationally time-consuming and difficult to make, especially near CEPs. The robustness of flash calculations depends on the formulation and the solution algorithm. One important part of the solution algorithm is the Rachford-Rice (RR) iteration. Michelsen (1994) first proposed an algorithm to solve the multiphase RR equations as a minimization of a convex objective function. Okuno et al. (2010a) developed the multiphase RR algorithm as a

2015 SPE Journal

1 ID: jaganm Time: 19:11 I Path: S:/J###/Vol00000/150050/Comp/APPFile/SA-J###150050

J169150 DOI: 10.2118/169150-PA Date: 21-July-15

CO2 solvent

UCEP: L1-L2 = V L1-V

LCEP: L1 = L2-V

L1-L2-V

Vaporization of L2 into V

L1-V

Oil

Extraction of oil components by L2

Fig. 1—CEP behavior observed in 1D compositional simulation when three-hydrocarbon-phase flow exists (after Okuno et al. 2011).

minimization of a nonmonotonic convex function with Nc linear constraints. Their RR method is guaranteed to converge to the correct phase split (if one exists) because no poles are within the feasible region defined by the linear constraints. The Okuno et al. (2010a) method is applied as part of the developed three-phase mixing-cell code in this paper. An important step in three-phase flash calculations is stability analysis. Stability analysis is performed for a single-phase mixture or individual phases of a multiphase mixture (Michelsen 1982; Li and Firoozabadi 2012). Good initial estimates are necessary for convergence especially for three-phase calculations because improper K-value estimates may not have a solution at all. Typically, initial estimates of K-values are available from the stability analysis of a two-phase mixture or, better yet, from the previous timestep in compositional simulation in which convergence of a three-phase flash was already obtained (Mohebbinia 2013). Previous K-value estimates in simulation are not necessarily available, however, when the number of equilibrium phases changes (Okuno et al. 2010b). The procedure for determining how many phases form is also clouded by the possibility of finding a false two-phase solution that is used for subsequent stability analysis. There are various experiments and computational methods for estimation of minimum miscibility pressure (MMP) for two-phase displacements (Jarrell et al. 2002). Slimtube experiments are the standard and best method to estimate MMP because they use real fluids. Slimtube experiments, however, are time-consuming and expensive, and one can perform in practice only a few MMP measurements for a fixed gas and oil composition. The experimental results can be sensitive to the packing material used and asphaltene precipitation (Elsharkawy et al. 1996). For some oil displacements, a “knee” in the recovery curve is also not always evident because of dispersion (Johns et al. 2002). For three-phase displacements, the recovery curve can bend gradually or abruptly with pressure or gas enrichment (Bhambri and Mohanty 2008; Okuno et al. 2011; Pedersen et al. 2012). Ahmadi and Johns (2011a) developed a simple and practical two-phase multiple-mixing-cell method with an Excel spreadsheet (PennPVT Toolkit 2010) to determine the MMP for displacements with any number of components. In this approach, all contacts are retained whether they are forward, backward, or in between. The two-phase multiple mixing-cell method was used to explore the mechanism of multiple-contact miscibility by tracking the tie lines that form in the cell contacts. The MMP is the pressure at which the first tie line in any cell becomes a critical tie-line. Ahmadi and Johns (2011) examined various ways to order and mix fluids cell to cell and showed that the proper way to mix them is to move the equilibrium phase most closely associated with the injection gas (typically, the vapor phase) forward of the other equilibrium phase (typically, the liquid phase). The two-phase mixing-cell method for MMP estimation was used to estimate MMPs for complex reservoir fluids and has compared well to MMP measurements from slimtubes. Typical MMP calculations take less than one minute, so one can perform many MMPs for a variety of initial oil and gas compositions (Ahmadi and Johns 2011). The MMP estimate from a mixing cell is reliable on the basis of good fluid characterizations because it is based on a cubic equation of state (EOS). Egwuenu et al. (2008) showed that one should match the MMP estimated with analytical methods (or mixing-cell methods) to available slimtube MMPs before performing compositional simulation. Rezaveisi

Stage:

Page: 2

Total Pages: 11

et al. (2014) used the multiple-mixing-cell method to determine tie-lines for improvement in computational time and robustness of two-phase flash calculations in compositional simulation. They demonstrated that the computational efficiency of several phase-behavior-calculation methods on the basis of the multiple mixing-cell tie-lines is comparable to that of other state-of-theart techniques in an implicit-pressure/explicit-composition-type reservoir simulator (Rezaveisi et al. 2015). Besides mixing-cell models, analytical solutions of 1D displacements with the method of characteristics (MOC) are used frequently in two-phase displacements to calculate the MMP (Buckley and Leverett 1942; Helfferich and Klein 1970; Helfferich 1981; Dumore et al. 1984; Orr 2007). The MOC can estimate MMPs very quickly and accurately only when the fluid characterization with a cubic equation of state is reliable. The displacement mechanism and miscibility development for two-phase displacements are well-known (Orr et al. 1993; Johns et al. 1993) and confirmed by experimental results and numerous applications. The approach for MMP estimation of multicomponent displacements is currently based on Johns and Orr (1996), which developed a graphical method to calculate analytically key tie lines for an 11component displacement. The MOC approach was improved by Wang and Orr (1997) and Jessen et al. (1998). The analytical solution for injection of a mixture of gas, however, can be complicated by the existence of multiple tie-lines that satisfy the geometric construction (Yuan and Johns 2005; Ahmadi et al. 2011b). Further, the analytical solutions for two-phase displacements with complex phase behavior that bifurcates into L1-L2 and L1-V regions (similar to three-phase systems) have not yet been extended to displacements with more than three components. Khorsandi et al. (2014) showed that, for such complex ternary displacements, the MMP does not exist for some oil and gas compositions. In those cases, L1-L2 behavior exists for all higher pressures even though the displacement efficiency is greater than 95% so that effectively a pressure is reached at which high displacement efficiency occurs. LaForce and Johns (2005) developed analytical solutions with MOC for three-phase partially miscible flow in ternary systems. LaForce et al. (2008) extended the solutions to partially miscible four-component displacements, and later confirmed the validity of the analytical solutions by comparison to experimental results (LaForce et al. 2010). There are no analytical solutions, however, for three-phase displacements with four or more components. MOC solutions therefore are currently reliable for simpler two-phase systems, whereas mixing-cell models were shown to match well the MMPs from slimtube experiments with a variety of complex phase behavior. This does not negate the value of MOC theory, but it serves to underscore the more practical nature of the use of mixing-cell models as a more robust method of MMP estimation. In this paper, we develop a three-phase multiple-mixing-cell method to estimate the MMP and/or the pressure at which displacement efficiency is maximized for multicomponent displacements. In the first section, we describe our mixing-cell algorithm in detail, and define the various tie-line lengths needed for twoand three-phase regions. We then demonstrate how the MMP and/ or pressure for maximum displacement efficiency are/is determined from these tie-line lengths. We also compare the mixingcell results to 1D numerical simulations, showing excellent agreement for the fluid displacements examined. Multiple Mixing-Cell Model for Three-Phase Displacements The approach taken in developing the new mixing-cell model is built on the principles from the two-phase mixing-cell method by Ahmadi and Johns (2011), which relies on repeated contacts of neighboring cells and finding the pressure at which one of the tielines (extrapolated to an infinite number of contacts) in any cell becomes zero length. The trick in developing the new three-phase mixing model is to identify the equilibrium phase that moves forward to contact fluid in other cells, the phase that stays behind to be contacted in a backward contact, and the phase that remains in-

2

2015 SPE Journal ID: jaganm Time: 19:11 I Path: S:/J###/Vol00000/150050/Comp/APPFile/SA-J###150050

J169150 DOI: 10.2118/169150-PA Date: 21-July-15

G S G

Page: 3

Total Pages: 11

S

1st contact

O I

X

Stage:

F Y

2nd contact

O

TLIS S G

X

I

F Y

S

F

X

Y

O

TLFS

3rd contact

pseudophases

I

TLFI

Fig. 2—Ordering of equilibrium phases in the new three-phase multiple-mixing-cell method. G: injection-gas composition; O: initial oil composition; S: equilibrium slow-phase composition; I: equilibrium intermediate-phase composition; F: equilibrium fast-phase composition. Flow upstream is to the left. Two pseudophases (shown by X and Y) for each tie-triangle are formed before the next contact.

between. We typically label the phases that form in compositional simulation as L1 (oleic), L2 (CO2-rich liquid), and V (vapor), but the mixing-cell algorithm, as will be shown, is not dependent on that labeling. For only L-V phases, it is relatively easy to identify the phase types in either simulation or a mixing-cell model because the vapor phase, which typically contains more of the injected gas, moves ahead of the equilibrium liquid phase. For three-hydrocarbon phases, however, it is not trivial to determine the phase types and ordering of those phases because of the increased combinations that these cell-to-cell contacts can take, depending on the pressure. In simulation, for example, one should label the injected gas at higher pressure as the second liquid phase (L2), whereas for lower pressure, one should call it a vapor phase (V). If that labeling is not performed correctly in simulation, the recoveries will be in error and the simulation can fail because of saturation discontinuities as phases flip labels. The first contact in our mixing-cell model begins by mixing oil and gas in a specified ratio (50:50 in this paper) and then flashing the overall mixture that results. Two or three equilibrium phases may result from this first contact or in later contacts. The equilibrium phases and the initial gas and oil compositions are then ordered from upstream to downstream before the next contact on the basis solely of their similarity to the initial oil and injection-gas compositions (Fig. 2). No matter how many equilibrium phases form in one cell, the equilibrium phase with a composition that is nearest the injection gas is moved ahead (downstream) of the other phase(s) and is labeled “fast” or F in Fig. 2. The equilibrium phase with a composition that is nearest the initial oil composition is placed in the rear and is labeled “slow” or S. If a third equilibrium phase exists, it is placed between these two phases and is labeled “intermediate” or I in Fig. 2. By default, if only two phases form, they are labeled either F or S. The phase closest to the oil and gas compositions is determined by the compositional distance between those compositions (say, compositions 1 and 2), which is given by kz1i  z2i k. The ordering of the phases is critical to the determination of minimum miscibility pressure (MMP) because it determines which compositions (cells) that are not in equilibrium are mixed in the next contact. We examined variations in the ordering for the three-phase mixing-cell method and ultimately found that the correct ordering is related to the phase compositions, not their phase types. The correct ordering allows the injection-gas components to advance faster than those in the oil (Li 2013). The nature of the phase, whether we label it L1, L2, or V, is immaterial to the ordering that is determined by phase compositions alone. Often, the L1, L2, and V phases are the “slow,” “intermediate,” and “fast” phases, respectively, but this is not always so, especially at higher pressure where the L2 phase may be the fast phase. After the phases are properly ordered, the next contact is made. In the two-phase mixing-cell model of Ahmadi and Johns (2011),

F

Fig. 3—Schematic of a tie-triangle showing the construction of two pseudophases along the boundary of a tie-triangle. These pseudophase compositions are used in subsequent contacts between cells.

each phase is defined as a cell; however, this approach causes problems for a three-phase mixing-cell method. Whenever a third phase appears, the middle phase is in equilibrium with its neighboring phases and therefore cannot alter the compositions that result from mixing cells in the next contact. Further, if we mix only the fast phase with the cell ahead, or the slow phase with the cell behind, then three phases do not typically remain and the cell-to-cell contacts can oscillate between two and three phases. One way to potentially circumvent this is by performing a negative three-phase flash, but this adds complexity, and in this paper, we perform only negative two-phase-flash and positive three-phase-flash calculations. The approach taken allows for a continuous three-phase region by creating two pseudophases from the three equilibrium phases before the next contact. The pseudophases are created by mixing the slow and intermediate phases in a 50:50 mole fraction ratio, and the fast and intermediate phases also in a 50:50 ratio. This generates two pseudophases with compositions that lie on two tie-lines that bound the three-phase tie-triangle (Fig. 3). We use only these pseudophases in mixing the cells at the next contact, where the pseudophase from the slow (S) and intermediate phases (I) are mixed with the cell behind, whereas the pseudophases from the fast (F) and intermediate (I) phases are then mixed with the cell ahead. The methodology then follows similarly to the two-phase multiple mixing model developed in Ahmadi and Johns (2011). The estimation of MMP or high-displacement pressure (PHD) is found by finding the pressure at which any one of the two-phase tie lines or the tie lines associated with the sides of a tie triangle become zero length as pressure is increased. As we will show, an MMP is not always obtained when a third phase develops. Instead, one can reach a pressure at which displacement efficiency is high and one of the critical endpoints becomes zero length, whereas the second critical endpoint (CEP) is near-critical. If the two CEPs are obtained at the same pressure or in the same cell, the MMP is indeed reached. No matter the case, our mixing-cell model can identify all possibilities to reasonable accuracy. The two-phase mixing-cell algorithm by Ahmadi and Johns (2011) was modified to recognize how many critical endpoints are obtained by calculating the lengths of the tie-lines along the tietriangle boundaries. If only two phases are present in all cells, only one critical point is necessary to be reached at the MMP, whereas for three-phase displacements, we need to consider two tie-line lengths. One must determine these tie-line lengths by extrapolation to an infinite number of contacts so that the MMP or PHD is less sensitive to dispersion. This idea is similar to that used for simulation at which one can extrapolate the recoveries at, say, 1.2 pore volumes injected to an infinite number of gridblocks. The steps in the mixing-cell algorithm are summarized here: 1. Label the reservoir oil (xO ) slow (S) and injection gas (yG ) fast (F), as illustrated in Fig. 2. 2. Mix the upstream F phase with the downstream S phase with z ¼ xo þ aðyG  xo Þ. The mixing ratio can be smaller

2015 SPE Journal

3 ID: jaganm Time: 19:11 I Path: S:/J###/Vol00000/150050/Comp/APPFile/SA-J###150050

J169150 DOI: 10.2118/169150-PA Date: 21-July-15

3.

4.

5.

6.

7.

8.

or larger than 0.5, depending on the tie-lines that one wants to develop quicker. We typically use a ¼ 0.5 for contacts between cells in which the number of phases was the same in the previous contact, but 0.15 was used where two-phase cells transition to three-phase cells. The initial pressure for this flash calculation should be well below any expected MMP or pressure for high displacement efficiency. Perform a negative two-phase flash with various initial Kvalue guesses for all cells including K-values in a previous contact for the cells nearby the current cell in the sequence. No stability analysis is needed before this flash because a two-phase negative flash is always performed, although one should check if the two-phase state that results is not a false two-phase solution (yields the lowest overall Gibbs energy). If a previous flash converged to three phases in cells before and after the current cell, one can proceed directly to Step 5 with those K-values as the initial guess for the next contact. It may be possible with very heavy oils that a fourth-hydrocarbon phase can also form (Okuno and Xu 2014b). In that case, one could extend the mixing-cell algorithm by performing a stability analysis of the three-phase equilibrium solution and allowing for two intermediate phases. Four phases are not considered here. Perform a stability analysis on the resulting two phases to identify if the two phases should split to three hydrocarbon phases with the methodology in Michelsen (1982). If the two phases are stable, then go to Step 6. Perform a three-phase flash (Okuno et al. 2010b) with a variety of initial K-value guesses for use in Rachford-Rice iterations. It could be that the two-phase flash in Step 3 did not find the smallest Gibbs energy state for two phases. That is, there are several false two-phase solutions that a flash calculation can converge to, and one must try multiple guesses to have more confidence that these false twophase solutions are avoided. Order the equilibrium phases that result as fast (F), intermediate (I), and slow (S) when three hydrocarbon phases are present, and fast (F) or slow (S) when only two phases are present. These phases are identified with the compositional distance to the initial oil and injection-gas composition, as described previously. In most cases, the fast phase will be the vapor phase (V), the intermediate phase will be the CO2-rich second liquid phase (L2), and the slow phase will be the oleic phase (L1). At higher pressure, however, these phase labels may not be in this order so one should use F, I, and S instead. Calculate the tie-line lengths (for two- and three-phase regions) to identify if miscibility or high displacement efficiency was reached. The three-phase region consists of an infinite number of tie-triangles. Fig. 3 demonstrates sides of one tie-triangle. Tie-line lengths between the slow and fast phases, fast and intermediate phases, and the slow and intermediate phases are calculated by TLFS ¼ kxFi  xSi k, TLFI ¼ kxFi  xIi k, and TLIS ¼ kxIi  xSi k. Here, xFi , xIi , xSi represent equilibrium fast, intermediate, and slow compositions of the tie-triangle phases, respectively. Because the phase ordering is based on the initial oil and gas compositions, the tie-line length between the F and S phases (TLFS) will often correspond to the longest side of the three-phase tie triangle. TLFS can only approach zero length (intersect a critical point) with pressure when multicontact-miscibility is developed in the standard way as a two-phase displacement. Extrapolate the tie-line lengths on the basis of (1/N)m to an infinite number of contacts where N ¼ 1. The parameter m in this paper is determined by a best fit of the tie-line lengths to the last 30 cells, where 100 contacts were performed at a given pressure. The extrapolated tie-line lengths represent what would have occurred in these tieline lengths had the contacts continued. Better accuracy in

Stage:

Page: 4

Total Pages: 11

these extrapolations can be obtained by performing more contacts, but we found that 100 contacts are more than sufficient. The exponent m was typically in the range of 0.2 to 0.5, which is similar to what was observed by Ahmadi et al. (2011) for the two-phase mixing-cell method. 9. Check if miscibility is reached by determining the minimum tie-line lengths in all cells at the current contact level. If only two phases form in all cells, the MMP is the pressure when the extrapolated TLFS becomes zero. For three phases, the MMP is reached when both TLFI and TLIS become zero at any cell location as pressure is increased, which corresponds to both CEPs becoming zero length. One of the CEPs is likely to become zero length first. The pressure at which one of the CEPs becomes zero length is the pressure at which displacement efficiency may be high. The displacement efficiency at this pressure will be high (termed PHD in this paper) when the other tieline length on the boundary of the tie-triangle has K-values close to unity (is near the other CEP). 10. If miscibility or near miscibility is not reached, make a new contact at a new trial pressure. One could determine the new trial pressure on the basis of the approach by Ahmadi et al. (2011) to minimize the total computational time at which bisection is used on the basis of the extrapolated tie-line lengths becoming negative or positive. In this paper, we do not perform the bisection, but simply plot all contacts with pressure to illustrate the change in tie-line lengths with pressure. The new contacts are based solely on two-phase or pseudo-two-phase cell compositions. That is, if three phases formed in mixing two cells from the last contact, determine the compositions of the pseudo-phases, as illustrated in Fig. 3, for a tie triangle. The fast (F) pseudophase is the one with a composition that is formed from the F and I phases of the tie triangle, whereas the slow (S) pseudo-phase is formed from the I and S phases of the tie triangle. Mix all cells on the basis of the fast and slow phases with z ¼ xF þ aðyS  xF Þ and a value of a as described in Step 2. We only use the pseudophase compositions for mixing of cells if three phases are present. 11. Repeat Steps 3 through 10 for all cells until either the MMP or PHD is obtained from the extrapolated tie-line lengths. The total number of cells (with three phases combined to form pseudophases) is 2N þ 2, where N is the contact level. The previously described procedure is independent of phase labeling and fractional flow (three-phase relative permeabilities), which makes it very robust compared with performing detailed 1D three-phase compositional simulation. We have examined the impact of the mixing ratio a on the results and found that the results do not vary significantly with the value of this ratio, although certain tie-lines are developed faster or slower with the number of contacts, similarly to what was seen by Ahmadi and Johns (2011). Example Calculations In this section, several examples of efficient three-phase displacements are shown. We compare the tie-lines and their lengths from the mixing-cell calculations to those from detailed compositional simulation with UTCOMP (Chang et al. 1990). The oil recoveries are also calculated with UTCOMP. The results from UTCOMP are based on phase labeling for input into the three-phase relative permeability model (Corey’s model). Phase labeling, whether L1, L2, or V, requires a careful analysis of the phase densities so that the phase types are relatively continuous (spatially) in UTCOMP. In the figures, we use the phase labels from the simulation, whereas the mixing-cell results are identified only according to their relative ordering (F, I, or S). Unless otherwise stated, the simulation results are at 1,000 gridblocks, whereas the number of total contacts in the mixing cell varies from 50 to 1,000. The UTCOMP simulations are conducted at nearly constant pressure, with only a 10-psia change from the injector to producer. The

4

2015 SPE Journal ID: jaganm Time: 19:11 I Path: S:/J###/Vol00000/150050/Comp/APPFile/SA-J###150050

J169150 DOI: 10.2118/169150-PA Date: 21-July-15

Stage:

Page: 5

Total Pages: 11

ω

Table 1—Composition of oil and gas and fluid properties for Case 1 (from Okuno et al. 2010b).

ω

Table 2—Composition of oil and gas and fluid properties for Case 2 (from Khan et al. 1992).

ω

Table 3—Composition of oil and gas and fluid properties for Case 3 (from Lim et al. 1992).

profiles from the simulations are always generated at 0.5 hydrocarbon pore volumes injected (HCPVI), but the recoveries are at 2.0 HCPVI. Both tie-line lengths and recoveries from UTCOMP are extrapolated to an infinite Pe´clet number (an infinite number of gridblocks) by performing a best fit to the values obtained with 50, 100, 200, 500, and 1,000 gridblocks. The best fits are obtained by plotting the tie-line lengths and recoveries as a function of ð1=Npe Þm . Each curve is fit separately. In some cases, the extrapo-

Tie-Line Length

1

0.1

50 Contacts TLFS 50 Contacts TLFI 50 Contacts TLIS 1,000 Contacts TLFS 1,000 Contacts TLFI 1,000 Contacts TLIS

0.01

0.001 0

0.2

0.4

0.6

0.8

1

Normalized Cell Number Fig. 4—Comparison of tie-line lengths for 50 and 1,000 contacts with the three-phase mixing cell for the four-component displacement (Case 1) at 1,200 psi and 105º F.

lated values were based solely on 200, 500, and 1,000 gridblocks if the R2 value was not close to 1.0. The first case considered is the quaternary displacement of Bob Slaughter Block-Q (BSB-Q) oil by CO2 solvent (95% CO2 and 5% C1) at the reservoir temperature of 105  F. The BSB-Q fluid model is given in Okuno et al. (2010b), and is based on the original BSB oil that consists of seven components. The PengRobinson equation-of-state (EOS) (Peng and Robinson 1976) parameters for the BSB-Q oil are given in Table 1. The second reservoir oil is the BSB west Texas oil (Khan et al. 1992) at a temperature of 105  F. The fluid properties for BSB are summarized in Table 2. The third reservoir fluid is from Monahans Clearfork (MC) (Lim et al. 1992), which is a low-temperature reservoir (90  F) from west Texas. Fluid properties for the MC oil are shown in Table 3. We first demonstrate the development of tie-lines from the mixing-cell model for Case 1 with the BSB-Q oil. As shown in Fig. 4, the tie-line lengths reach approximately the same values for 50 and 1,000 contacts, but are better developed for 1,000 contacts. The x-axis was normalized in the figure by the number of contacts so that the curves would overlie each other. Fig. 5 shows a side-by-side comparison of tie-line lengths from the mixing cell (1,000 contacts) and 1D simulation (1,000 gridblocks) for Case 1 at 1,200 psia. The tie-line lengths appear to agree well, except for a small portion of the three-phase region developed. These differences occur because the locations of the tie-lines in the simulation are dictated by fractional flow, and the development of the tie-lines varies according to the level of mixing in the mixing-cell model and UTCOMP. The mixing-cell

2015 SPE Journal

5 ID: jaganm Time: 19:11 I Path: S:/J###/Vol00000/150050/Comp/APPFile/SA-J###150050

J169150 DOI: 10.2118/169150-PA Date: 21-July-15

Stage:

1

Page: 6

Total Pages: 11

1 TLL1-V

TLFS

Tie-Line Length

Tie-Line Length

TLIS 0.1 TLFI 0.01

0.001

TLL1-L2

0.1

TLL2-V 0.01

0.001 0

200

400

600

800

1,000

0

0.4

0.2

Cell Number

0.6

Dimensionless Distance

Fig. 5—Comparison of tie-line lengths generated for the four-component displacement (Case 1) at 1,200 psia and 105º F. Left: mixing-cell model (1,000 contacts). Right: 1D simulation (1,000 gridblocks).

1

1 TLL1-V

0.1

0.01

Tie-Line Length

Tie-Line Length

TLFS TLIS

TLFI

0.001

TLL1-L2

0.1

0.01 TLL2-V 0.001

0

200

400

600

800

0

1,000

0.2

0.4

0.6

Dimensionless Distance

Cell Number

Fig. 6—Comparison of tie-line lengths generated for the four-component displacement (Case 1) at 1,300 psia and 105º F. Left: mixing-cell model (1,000 contacts). Right: 1D simulation (1,000 gridblocks).

results do not include fractional flow effects so that there is no position or time scale associated with them. Fig. 6 shows a similar side-by-side comparison at a higher pressure of 1,300 psia. Figs. 7 and 8 illustrate a better way to compare the simulation results to the mixing-cell tie-line lengths. In these figures, the tielines of displacement at 1,200 and 1,300 psia are compared in tiesimplex space (!-space). !-space reduces the dimensionality of the displacement and eliminates time and distance from the simu¼ lation results (Voskov and Tchelepi 2008). In this figure c12 1 ¼ ðx14 þ x24 Þ=2, where we chose Component 1 ðx11 þ x21 Þ=2 and c12 4 to be the lightest component and Component 4 the heaviest. One point in !-space for a quaternary displacement therefore correand c12 sponds to one tie-line as defined by c12 4 . As shown, the 1 0.3

0.3

Simulation Mixing-cell FS tie Line Mixing-cell FI tie Line Mixing-cell IS tie Line

Simulation Mixing-cell FS tie Line Mixing-cell FI tie Line Mixing-cell IS tie Line

Injection tie line 0.2

Injection tie line L1V

γ4

γ4

0.2

agreement between all phases and tie-line lengths is remarkable, although some portions of the curves are better developed in either the mixing-cell model or the simulation. As stated previously, one can alter the development of various tie-lines by changing the mixing ratio a. These plots demonstrate that the procedure to order the phases in the mixing-cell model is correct. Figs. 9 and 10 compare the mixing-cell tie lines at 1,350 psia and 1,000 contacts when the ordering of the phases is reversed. That is, in Fig. 9, the ordering is incorrect in that the phase with a composition that is nearest the injection-gas composition (normally the F phase) is placed between the I and S phases. Fig. 10 shows the correct ordering in which the F phase is downstream of the I and S phases. In UTCOMP, Fig. 10 corresponds to injection

L1V L1L2

0.1 Initial tie line

L1L2

0.1

L1V

L 1V Initial tie line

L2V

0

L2V

0 0

0.2

0.6

0.4

0.8

1

γ1 Fig. 7—Tie-simplex comparison of composition path in Fig. 5 between three-phase mixing-cell model and 1D simulation for the four-component displacement (Case 1) at 1,200 psia and 105º F.

0

0.2

0.4

0.6

0.8

1

γ1 Fig. 8—Tie-simplex comparison of composition path in Fig. 6 between three-phase mixing-cell model and 1D simulation for the four-component displacement (Case 1) at 1,300 psia and 105º F.

6

2015 SPE Journal ID: jaganm Time: 19:11 I Path: S:/J###/Vol00000/150050/Comp/APPFile/SA-J###150050

J169150 DOI: 10.2118/169150-PA Date: 21-July-15

Stage:

1

Total Pages: 11

1 TLIS

TLFS

Tie-Line Length

TLFS

Tie-Line Length

Page: 7

0.1 TLFI

0.01

TLIS

0.1

TLFI 0.01

0.001

0.001 0

200

400

600

800

0

1,000

200

Cell Number

400

600

800

1,000

Cell Number

Fig. 9—Tie-line length development for the mixing-cell model with 1,000 contacts at 1,350 psia and 105º F assuming the wrong phase ordering. The phase that is second-closest in composition to the injection gas is placed as the fast phase. This would correspond to the injection gas being labeled vapor at this pressure.

Fig. 10—Tie-line length development for the mixing-cell model with 1,000 contacts at 1,350 psia and 105º F assuming the correct ordering. The phase that is closest in composition to the injection gas is placed as the fast phase. This would correspond to the injection gas being labeled L2.

1.0E+6

1.0E+6

1.0E+4

1.0E+4 PC2

1.0E+2

1.0E+2

CO2

1.0E+0

CO2

KIS

KFI

PC1

C1

1.0E+0

C1 1.0E–2

1.0E–2

1.0E–4

1.0E–4 200

0

400

600

800

1,000

Cell Number

PC1 PC2 0

200

400

600

800

1,000

Cell Number

Fig. 11—KFI-values from the mixing-cell model with 1,000 contacts for the four-component displacement (Case 1) at 1,300 psia and 105º F.

Fig. 12—KIS-values from the mixing-cell model with 1,000 contacts for the four-component displacement (Case 1) at 1,300 psia and 105º F.

of the second-liquid phase (L2), whereas Fig. 9 corresponds to assuming that the injection gas is vapor (V). At this higher pressure of 1,350 psia, the densities of the V and L2 phases invert compared with the case at 1,300 psia. Density inversion is wellknown, but is difficult to handle in simulation in which phases are labeled. One must change the threshold phase densities between the vapor phase and the CO2-rich second-liquid phase by trialand-error. One also can use the values of the equilibrium ratios (K-values) to examine mass transfer between phases. Figs. 11 and 12

show two sets of K-values for the three-phase region at 1,300 psia, where KiFI ¼ xFi =xIi and KiIS ¼ xIi =xSi . Three-phase regions are shaded in blue, and two-phase regions are shaded in gray. The intermediate phase (L2 phase at this pressure) aids in the extraction of heavy components from the slow phase (L1 phase) to the fast phase (V phase at this pressure). K-values on the upstream and downstream edge of three-phase region approach unity. That is, the intermediate phase (I phase) serves as a buffer between the F and S phases. The F and S two-phase region ahead of the threephase region exhibits condensing behavior, whereas behind the three-phase region, the F and S phases show vaporizing behavior. Fig. 13 gives the minimum mixing-cell tie-line lengths between the F and I phases as a function of the number of contacts (1/Nm) at a pressure of 1,200 psia, resulting from the first to 100th contact. The solver in Excel is used to optimize the exponent m to give the best fit for the last 30 data points (from the 71st to 100th contact). Fig. 14 shows the extrapolated tie-line lengths and recoveries as a function of pressure from the mixing-cell model and UTCOMP simulations. As is shown, the extrapolated recovery from simulation reaches 98% at 1,300 psia. This corresponds to near miscibility in which the composition path goes through a critical endpoint indicated by a negative extrapolated tie-line length for the L2-V phases. Extrapolated tie-line lengths are negative when the dispersion-free path goes through a critical point. Negative tie-line lengths were used by Ahmadi et al. (2011) to perform bisection until the pressure at which the extrapolated tie-line length that is zero is reached [normally the minimum miscibility pressure (MMP)]. The mixing-cell results also show a zero-extrapolated tie-line length between the F and I phases at 1,300 psia. We label this high-displacement pressure (PHD),

Extrapolated Minimum Tie-Line Length Between F and I

0.2

0.15

0.1

0.05

0

0

0.2

0.4

0.6

0.8

1

(1/N)m Fig. 13—Extrapolation of the minimum tie-line length between fast and intermediate phases at a pressure of 1,200 psia to determine TL‘ FI . A value of m 5 0.25 gives the best fit, and corresponds to TL‘ FI 5 0.072. 2015 SPE Journal

7 ID: jaganm Time: 19:11 I Path: S:/J###/Vol00000/150050/Comp/APPFile/SA-J###150050

Stage:

0.6

0.6

0.5

0.5

0.4

Extrapolated Tie Line

Extrapolated Tie Line

J169150 DOI: 10.2118/169150-PA Date: 21-July-15

TLFS

0.3 0.2

TLIS

0.1 TLFI

0 –0.1 800

PHD 1,000

1,200

MMP 1,400

1,600

1,800

Total Pages: 11

1

RF

0.8 0.4

TLL1-V 0.6

0.3 0.2

TLL1-L2

0.4

0.1 0 –0.1 800

2,000

Page: 8

TLL2-V

0.2 0

1,000 1,200 1,400 1,600 1,800 2,000

Pressure (psia)

Pressure (psia)

Fig. 14—Comparison of the extrapolated tie-line lengths and extrapolated recoveries for Case 1. Left: mixing-cell model (100 contacts). Right: 1D simulation (1,000 gridblocks).

1

0.6

0.6

RF 0.5

Extrapolated Tie Line

Extrapolated Tie Line

0.5 0.4 TLFS

TLIS

0.3 0.2 0.1

TLFI

0 –0.1 800

PHD 1,000

1,200

1,400

0.8

0.4 0.6

0.3 TLL1-L2

TLL1-V

0.2

0.4

0.1

TLL2-V 0.2

0 –0.1

1,600

1,800

Pressure (psia)

800

1,000

1,200

1,400

1,600

0 1,800

Pressure (psia)

Fig. 15—Comparison of the extrapolated tie-line lengths and extrapolated recoveries for Case 2. Left: mixing-cell model (100 contacts). Right: 1D simulation (1,000 gridblocks).

which is the pressure for high displacement efficiency. High displacement efficiency is possible when both critical endpoints (CEPs) (upper and lower) are reached or nearly reached during the displacement. In this case, the CEP between the F and I phases is reached, whereas the tie-line length between the I and S phases is relatively small, indicating K-values that are close to unity. The simulation results show similar features, although the CEP between the I and S phases (L1 and L2 here) is even smaller and is nearly zero. These differences are likely the result of the mixingcell model with only 100 contacts, whereas the simulations were performed with up to 1,000 gridblocks. It is not clear if multicontact miscibility (MCM) is reached in this displacement at 1,300 psia, but the mixing-cell and simulation results both indicate that one should obtain high displacement efficiency. The tie-line lengths in Fig. 14 show unusual behavior at larger pressures than PHD. Both the simulation and mixing-cell tie-line lengths increase with pressure (and the simulation recoveries decrease slightly), but the mixing-cell tie-line length between the F and S phases quickly approaches zero length at 2,000 psia. This is clearly MCM behavior and so we have taken this to be the MMP. This behavior is anomalous because normally the tie-line between F and S should be longer than the other sides of the tie-triangle. However, this behavior could occur if phase behavior bifurcates (a critical point separates into two critical points), as demonstrated by Khorsandi et al. (2014). The simulation results are inconclusive above 1,600 psia because they were difficult to obtain at these pressures, and many simulations failed. The three-phase region disappeared in the simulations above a pressure of approximately 1,600 psia, and only liquid-liquid behavior remained (likely, L1 and L2). However, the simulations were difficult to perform at these pressures, and it is not clear that the flash calculations correctly identified the phases. That is, three phases may exist at these pressures,

but stability analysis failed to find them. It is also possible that the simulations, which have less mixing when 1,000 gridblocks are used, avoided the three-phase region compared with the mixingcell model that used only 100 total contacts. The extrapolated tie-line lengths from mixing cell are compared with the extrapolated tie-line lengths from simulation for Cases 2 and 3 and are compared in Figs. 15 and 16, respectively. These cases show similar features as those for Case 1. Case 3 indicates a true MMP at a pressure of approximately 1,450 psia, which is greater than the pressure for high displacement efficiency at approximately 1,150 psia. The MMP here is determined where TLFS is approximately zero. The results from the mixing-cell model and UTCOMP are compared in Table 4. We calculated further the MMP with PennPVT (2010), which is currently based on only allowing for two phases. The results between the three-phase UTCOMP simulations and the three-phase multiple-mixing-cell model agree well with the accuracy calculated. The MMPs from the two-phase mixing-cell model do not agree as closely with these results. Conclusions We developed a new three-phase multiple-mixing-cell method that can estimate the pressure at which high displacement efficiency occurs. We show how to order the cells on the basis of the compositional distance between their compositions and the initial oil and injection-gas compositions. The mixing-cell tie-line lengths when extrapolated to an infinite number of contacts agree well with detailed three-phase simulations also extrapolated to an infinite number of gridblocks. When plotted in tie-simplex space, the mixing-cell and simulation results encountered nearly identical tie-triangles and two-phase tie lines.

8

2015 SPE Journal ID: jaganm Time: 19:11 I Path: S:/J###/Vol00000/150050/Comp/APPFile/SA-J###150050

Stage:

0.6

0.6

0.5

0.5

0.4 TLFS 0.3 0.2 TLIS

0.1

MMP

0 TLFI

–0.1 800

PHD

1,000

1,200

1,400

1,600

Extrapolated Tie Line

Extrapolated Tie Line

J169150 DOI: 10.2118/169150-PA Date: 21-July-15

Page: 9

Total Pages: 11

1

RF

0.8

0.4 0.6

0.3

TLL1-V

0.2

0.4 TLL1-L2

0.1

0.2 0 –0.1 800

TLL2-V 1,000

Pressure (psia)

1,200

1,400

0 1,600

Pressure (psia)

Fig. 16—Comparison of the extrapolated tie-line lengths and extrapolated recoveries for Case 3. Left: mixing-cell model (100 contacts). Right: 1D simulation (1,000 gridblocks).

Table 4—Comparison of PHD and MMP.

High-displacement pressure occurs when one or more critical endpoints are reached, or nearly so, during the displacement. The mixing-cell approach is more robust than those from simulation because the phases are not labeled according to type, but rather they are ordered on the basis of how close their composition is to the initial oil and injection-gas composition. The mixing-cell model also does not use three-phase relative permeability, making it easier to implement; it also is significantly faster than simulation. Nomenclature F ¼ fast phase I ¼ intermediate phase K ¼ equilibrium ratio L1 ¼ oleic phase L2 ¼ solvent-rich liquid phase m ¼ extrapolation exponent N ¼ number of contacts Nc ¼ number of components Npe ¼ Pe´clet number Pc ¼ critical pressure PHD ¼ high-displacement pressure S ¼ slow phase Tc ¼ critical temperature TL ¼ tie-line length V ¼ vapor phase x ¼ phase composition y ¼ phase composition z ¼ overall composition x ¼ acentric factor c ¼ c-space parameter Subscripts F ¼ fast phase i ¼ component I ¼ intermediate phase L1 ¼ oleic phase L2 ¼ solvent-rich liquid phase

S ¼ slow phase V ¼ vapor phase Superscripts G ¼ gas O ¼ oil Acknowledgments The authors gratefully thank BP, Chevron, Denbury, KNPC, Maersk, OMV, and Shell for their financial support of this research through the Enhanced Oil Recovery Joint Industry Project in the EMS Energy Institute at the Pennsylvania State University at University Park, Pennsylvania. Russell T. Johns holds the Victor and Anna Mae Beghini Faculty Fellowship in Petroleum and Natural Gas Engineering at the Pennsylvania State University. He is also the Petroleum and Natural Gas Engineering Program Chair.

References Ahmadi, K. and Johns, R. T. 2011. Multiple Mixing-Cell Method for MMP Calculations. SPE J. 16 (4): 732–742. SPE-116823-PA. http:// dx.doi.org/10.2118/116823-PA. Ahmadi, K., Johns, R. T., Mogensen, K. et al. 2011. Limitations of Current Method-of-Characteristics (MOC) Methods Using Shock-Jump Approximations to Predict MMPs for Complex Gas/Oil Displacements. SPE J. 16 (4): 743–750. SPE-129709-PA. http://dx.doi.org/ 10.2118/129709-PA. Bhambri, P. and Mohanty, K. K. 2008. Two- and Three-Hydrocarbon Phase Streamline-Based Compositional Simulation of Gas Injection. J. Petrol. Sci. & Eng. 62 (1–2): 16–27. http://dx.doi.org/10.1016/ j.petrol.2008.06.03. Blunt, M. J. 2000. An Empirical Model for Three-Phase Relative Permeability. SPE J. 5 (4): 435–445. SPE-67950-PA. http://dx.doi.org/ 10.2118/67950-PA. Buckley, S. E. and Leverett, M. C. 1942. Mechanism of Fluid Displacement in Sands. Trans. AIME 146: 107–116. SPE-942107-G-PA. http:// dx.doi.org/10.2118/942107-G-PA.

2015 SPE Journal

9 ID: jaganm Time: 19:12 I Path: S:/J###/Vol00000/150050/Comp/APPFile/SA-J###150050

J169150 DOI: 10.2118/169150-PA Date: 21-July-15

Chang, Y-b, Pope, G. A., and Sepehrnoori, K. 1990. A Higher-Order Finite-Difference Compositional Simulator. J. Petrol. Sci. & Eng. 5 (1): 35–50. http://dx.doi.org/10.1016/0920-4105(90)90004-M. Delshad, M. and Pope, G. A. 1989. Comparison of the Three-Phase Oil Relative Permeability Models. Trans. Porous Med. 4 (1): 59–83. Dumore, J. M., Hagoort, J., and Risseeuw, A. S. 1984. An Analytical Model for One-Dimensional, Three-Component Condensing and Vaporizing Gas Drives. SPE J. 24 (2): 169–179. SPE-10069-PA. http://dx.doi.org/10.2118/10069-PA. Egwuenu, A. M., Johns, R. T., and Li, Y. 2008. Improved Fluid Characterization for Miscible Gas Floods. SPE Res Eval & Eng 11 (4): 655–665. SPE-94034-PA. http://dx.doi.org/10.2118/94034-PA. Elsharkawy, A. M., Poettmann, F. H., and Christiansen, R. L. 1996. Measuring CO2 Minimum Miscibility Pressures: Slim-tube or Rising-Bubble Method? Energy Fuels 10 (2): 443–449. http://dx.doi.org/10.1021/ ef940212f. Gardner, J. W., Orr, F. M. Jr., and Patel, P. D. 1981. The Effect of Phase Behavior on CO2-Flood Displacement Efficiency. J Pet Technol 33 (11): 2067–2081. SPE-8367-PA. http://dx.doi.org/10.2118/8367-PA. Griffiths, R. B. 1974. Thermodynamic Model for Tricritical Points in Ternary and Quaternary Fluid Mixtures. J. Chemical Physics 60 (1): 195–206. http://dx.doi.org/10.1063/1.1680768. Guler, B., Wang, P., Pope, G. A. et al. 2001. Three- and Four-Phase Flow Compositional Simulations of CO2/NGL EOR. Presented at the SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, USA, 30 September–3 October. SPE-71485-MS. http:// dx.doi.org/10.2118/71485-MS. Helfferich, F. G. 1981. Theory of Multicomponent, Multiphase Displacement in Porous Media. SPE J. 21 (1): 61–62. SPE-8372-PA. http:// dx.doi.org/10.2118/8372-PA. Helfferich, F. and Klein, G. 1970. Multicomponent Chromatography. New York City: Marcel Dekker Inc. Jarrell, P. M., Fox, C. E., Stein, M. H. et al. 2002. Practical Aspects of CO2 Flooding, SPE Monograph Series. Richardson Texas: Society of Petroleum Engineers. Jerauld, G. R. 1997. General Three-Phase Relative Permeability Model for Prudhoe Bay. SPE Res Eng 12 (4): 255–263. SPE-36178-PA. http:// dx.doi.org/10.2118/36178-PA. Jessen, K., Michelsen, M. L., and Stenby, E. H. 1998. Global Approach for Calculation of Minimum Miscibility Pressure. Fluid Phase Equilibria 153 (2): 251–263. http://dx.doi.org/10.1016/S0378-3812(98)00414-2. Johns, R. T., Dindoruk, B., and Orr, F. M. 1993. Analytical Theory of Combined Condensing/Vaporizing Gas Drive. SPE Advanced Technology Series 1 (2): 7–16. SPE-24112-PA. http://dx.doi.org/10.2118/24112-PA. Johns, R. T. and Orr, F. M. Jr. 1996. Miscible Gas Displacement of Multicomponent Oils. SPE J. 1 (1): 39–50. SPE-30798-PA. http:// dx.doi.org/10.2118/30798-PA. Johns, R. T., Sah, P., and Solano, R. 2002. Effect of Dispersion on Local Displacement Efficiency for Multicomponent Enriched-Gas Floods Above the Minimum Miscibility Enrichment. SPE Res Eval & Eng 5 (1): 4–10. SPE-64725-PA. http://dx.doi.org/10.2118/64725-PA. Juanes, R. and Patzek, T. W. 2004. Three-Phase Displacement Theory: An Improved Description of Relative Permeability. SPE J. 9 (3): 302–313. SPE-77539-PA. http://dx.doi.org/10.2118/77539-PA. Khan, S. A., Pope, G. A., and Sepehrnoori, K. 1992. Fluid Characterization of Three-Phase CO2/Oil Mixtures. Presented at the SPE/DOE Enhanced Oil Recovery Symposium, Tulsa, Oklahoma, USA, 22–24 April. SPE-24130-MS. http://dx.doi.org/10.2118/24130-MS. Khorsandi, S., Ahmadi, K., and Johns, R. T. 2014. Analytical Solutions for Gas Displacements With Bifurcating Phase Behavior. SPE J. 19 (5): 943–955. SPE-166487-PA. http://dx.doi.org/10.2118/166487-PA. LaForce, T., Cinar, Y., Johns, R. T. et al. 2010. Experimental Confirmation of Analytical Composition Routes in Three-Phase Partially Miscible Flow. SPE J. 15 (1): 160–170. SPE-99505-PA. http://dx.doi.org/ 10.2118/99505-PA. LaForce, T., Jessen, K., and Orr, F. M. Jr. 2008. Four-Component Gas/ Water/Oil Displacements in One Dimension: Part I, Structure of the Conservation Law. Trans. Porous Med. 71 (2): 199–216. http:// dx.doi.org/10.1007/s11242-007-9120-9. LaForce, T. and Johns, R. T. 2005. Composition Routes for Three-Phase Partially Miscible Flow in Ternary Systems. SPE J. 10 (2): 161–174. SPE-89438-PA. http://dx.doi.org/10.2118/89438-PA.

Stage:

Page: 10

Total Pages: 11

Li, L. 2013. Three-Phase Mixing Cell Method for Gas Flooding. MS thesis, the Pennsylvania State University, State College, Pennsylvania (May 2013). Li, Z. and Firoozabadi, A. 2012. General Strategy for Stability Testing and Phase-Split Calculation in Two and Three Phases. SPE J. 17 (4): 1096–1107. SPE-129844-PA. http://dx.doi.org/10.2118/129844-PA. Li, Y. K. and Nghiem, L. X. 1986. Phase Equilibria of Oil, Gas and Water/Brine Mixtures From a Cubic Equation of State and Henry’s Law. Canadian J. Chemical Eng. 64 (3): 486–496. http://dx.doi.org/ 10.1002/cjce.5450640319. Lim, M. T., Khan, S. A., Sepehrnoori, K. et al. 1992. Simulation of Carbon Dioxide Flooding Using Horizontal Wells. Presented at the SPE Annual Technical Conference and Exhibition, Washington, DC, USA, 4–7 October. SPE-24929-MS. http://dx.doi.org/10.2118/24929-MS. Lins, A. G. Jr., Nghiem, L. X., and Harding, T. G. 2011. Three-Phase Hydrocarbon Thermodynamics Liquid-Liquid-Vapour Equilibrium in CO2 Process. Presented at the SPE Reservoir Characterization and Simulation Conference and Exhibition, Abi Dhabi, UAE, 9–11 October. SPE-148040-MS. http://dx.doi.org/10.2118/148040-MS. Michelsen, M. L. 1982. The Isothermal Flash Problem. Part I. Stability. Fluid Phase Equilibria 9 (1): 1–19. http://dx.doi.org/10.1016/03783812(82)85001-2. Michelsen, M. L. 1994. Calculation of Multiphase Equilibrium. Computers & Chemical Eng. 18 (7): 545–550. http://dx.doi.org/10.1016/ 0098-1354(93)E0017-4. Mohebbinia, S. 2013. Advanced Equation of State Modeling for Compositional Simulation of Gas Floods. PhD dissertation, the University of Texas at Austin, Austin, Texas (December 2013). Okuno, R., Johns, R. T., and Sepehrnoori, K. 2010a. A New Algorithm for Rachford-Rice for Multiphase Compositional Simulation. SPE J. 15 (2): 313–325. SPE-117752-PA. http://dx.doi.org/10.2118/117752-PA. Okuno, R., Johns, R. T., and Sepehrnoori, K. 2010b. Three-Phase Flash in Compositional Simulation Using a Reduced Method. SPE J. 15 (3): 689–703. SPE-125226-PA. http://dx.doi.org/10.2118/125226-PA. Okuno, R., Johns, R. T., and Sepehrnoori, K. 2011. Mechanisms for High Displacement Efficiency of Low-Temperature CO2 Floods. SPE J. 16 (4): 751–767. SPE-129846-PA. http://dx.doi.org/10.2118/129846-PA. Okuno, R. and Xu, Z. 2014a. Mass Transfer on Multiphase Transitions in Low-Temperature Carbon Dioxide Floods. SPE J. 19 (6): 1005–1023. SPE-166345-PA. http://dx.doi.org/10.2118/166345-PA. Okuno, R. and Xu, Z. 2014b. Efficient Displacement of Heavy Oil by Use of Three Hydrocarbon Phases. SPE J. 19 (5): 956–973. SPE-165470PA. http://dx.doi.org/10.2118/165470-PA. Orr, F. M. Jr. 2007. Theory of Gas Injection Processes. Denmark: TieLine Publications. Orr, F. M. Jr., Johns, R. T., and Dindoruk, B. 1993. Development of Miscibility in Four-Component CO2 Floods. SPE Res Eng 8 (2): 135–142. SPE-22637-PA. http://dx.doi.org/10.2118/22637-PA. Orr, F. M. Jr., Silva, M. K., and Lien, C. L. 1983. Equilibrium Phase Compositions of CO2/Crude Oil Mixtures—Part 2: Comparison of Continuous Multiple-Contact and Slim-Tube Displacement Tests. SPE J. 23 (2): 281–291. SPE-10725-PA. http://dx.doi.org/10.2118/10725-PA. Pedersen, K. S., Leekumjorn, S., Krejbjerg, K. et al. 2012. Modeling of EOR PVT Data Using PC-SAFT Equation. Presented at the International Petroleum Exhibition and Conference, Abu Dhabi, UAE, 11–14 November. SPE-162346-PA. http://dx.doi.org/10.2118/162346-PA. Peng. D.-Y. and Robinson, D. B. 1976. A New Two-Constant Equation of State. Industrial & Eng. Chemistry Fundamentals 15 (1): 59–64. http://dx.doi.org/10.1021/i160057a011. PennPVT Toolkit. 2010. Gas Flooding Joint Industry Project. EMS Energy Institute, Pennsylvania State University, University Park, Pennsylvania. Perschke, D. R., Pope, G. A., and Sepehrnoori, K. 1989. Phase Identification During Compositional Simulation. In 1989 CEOGRR Annual Report, Category C, Section II. Rezaveisi, M., Johns, R. T., and Sepehrnoori, K. 2015. Application of Multiple Mixing-Cell Method to Improve Speed and Robustness of Compositional Simulation. SPE J. Preprint. Rezaveisi, M., Sepehrnoori, K., and Johns, R. T. 2014. Tie-Simplex-Based Phase-Behavior Modeling in an IMPEC Reservoir Simulator. SPE J. 19 (2): 327–339. SPE-163676-PA. http://dx.doi.org/10.2118/163676-PA.

10

2015 SPE Journal ID: jaganm Time: 19:12 I Path: S:/J###/Vol00000/150050/Comp/APPFile/SA-J###150050

J169150 DOI: 10.2118/169150-PA Date: 21-July-15

Voskov, D. and Tchelepi, H. A. 2008. Compositional Space Parametrization for Miscible Displacement Simulation. Trans. Porous Med. 75 (1): 111–128. http://dx.doi.org/10.1007/s11242-008-9212-1. Wang, Y. and Orr, F. M. Jr. 1997. Analytical Calculation of Minimum Miscibility Pressure. Fluid Phase Equilibria 139 (1): 101–124. http:// dx.doi.org/10.1016/S0378-3812(97)00179-9. Wang, X. and Sttrycker, A. 2000. Evaluation of CO2 Injection With Three Hydrocarbon Phases. Presented at the SPE International Oil and Gas Conference and Exhibition, Beijing, China, 7–10 November. SPE64723-MS. http://dx.doi.org/10.2118/64723-MS. Widom, B. 1973. Tricritical Points in Three- and Four-Component Fluid Mixtures. J. Phys. Chem. 77 (18): 2196–2200. http://dx.doi.org/ 10.1021/j100637a008, Yellig, W. F. and Metcalfe, R. S. 1980. Determination and Prediction of CO2 Minimum Miscibility Pressures. J Pet Technol 32 (1): 160–168. SPE-7477-PA. http://dx.doi.org/10.2118/7477-PA. Yuan, H. and Johns, R. T. 2005. Analytical Solutions for Gas Displacements With Bifurcating Phase Behavior. SPE J. 10 (4): 416–425. Yuan, C. and Pope, G. A. 2012. A New Method to Model Relative Permeability in Compositional Simulators to Avoid Discontinuous Changes Caused by Phase-Identification Problems. SPE J. 17 (4): 1221–1230. SPE-142093-PA. http://dx.doi.org/10.2118/142093-PA.

Liwei Li is a PhD degree candidate in the Petroleum and Natural Gas Engineering Program at Pennsylvania State University. Her research interests include compositional simulation and phase behavior of enhanced oil recovery (EOR). Li holds a BS degree in applied chemistry from China University of Mining and Technology (Beijing) and an MS degree in petroleum and natural gas engineering from Pennsylvania State University. Saeid Khorsandi is a PhD degree candidate in the Petroleum and Natural Gas Engineering Program at Pennsylvania State

Stage:

Page: 11

Total Pages: 11

University. His research interests include mathematics of EOR, phase-behavior calculation, and reservoir simulation. Khorsandi holds a BS degree in petroleum-production engineering from Petroleum University of Technology, Iran, and an MS degree in reservoir engineering from Sharif University of Technology, Iran. Russell T. Johns is the Petroleum and Natural Gas Engineering Program Chair within the Department of Energy and Mineral Engineering at the Pennsylvania State University. He also holds the Victor and Anna Mae Beghini Professorship of Petroleum and Natural Gas Engineering and the CMG Foundation Chair in fluid behavior and rock interactions. Before Johns’ current position, he served on the petroleum-engineering faculty at the University of Texas at Austin from 1995 to 2010. He also has 9 years of industrial experience as a petrophysical engineer with Shell Oil and as a consulting engineer for Colenco Power Consulting in Baden, Switzerland. Johns holds a BS degree in electrical engineering from Northwestern University and MS and PhD degrees in petroleum engineering from Stanford University. He has more than 200 publications in EOR, thermodynamics and phase behavior, unconventional gas engineering, multiphase flow in porous media, and well testing. Johns received the SPE Ferguson Medal in 1993 and served as coexecutive editor for SPE Reservoir Evaluation and Engineering from 2002 to 2004. In 2009, he was awarded SPE Distinguished Member status, and in 2013, he received the SPE Faculty Pipeline Award. Johns is currently director of the Enhanced Oil Recovery Consortium and codirector of the Unconventional Natural Resources Consortium in the EMS Energy Institute at Pennsylvania State University. Kaveh Ahmadi is a reservoir engineer with BP America. His current research interests are in well-performance prediction, pressure-transient analysis, and phase-behavior calculation. He holds a PhD degree in petroleum engineering from the University of Texas at Austin.

2015 SPE Journal

11 ID: jaganm Time: 19:12 I Path: S:/J###/Vol00000/150050/Comp/APPFile/SA-J###150050