Relative Permeability of Near-Miscible Fluids in Compositional Simulators

Abstract

Miscible gas injection is one of the most effective enhanced oil recovery techniques. There are several challenges in accurately modeling this process, which occurs in the near-miscible region. The adjustment of relative permeability for near-miscible processes is the main focus of this work. The dependence of relative permeability on phase identification can lead to significant complications while simulating near-miscible displacements. We present an analysis of how existing methods incorporate compositional dependence in relative permeability functions. The sensitivity of the different methods to the choice of reference points is presented with guidelines to limit the modification of the relative permeabilities to physically reasonable values. We distinguish between the two objectives of reflecting near-miscible behavior and ensuring smooth transitions across phase changes. We highlight an important link that combines the two objectives in a more general framework. We make use of Gibbs free energy as a compositional indicator in the generalized framework. The new approach was implemented in an automatic differentiation general purpose research simulator and tested on a set of near-miscible gas-injection problems. We show that including compositional dependencies in the relative permeability near the critical point impacts the simulation results with significant improvements in nonlinear convergence.

Appendix

This section includes additional 1D 3 component cases to highlight the behavior of models that focus on reflecting miscibility only. A 2D four-component case is also presented to provide additional comparison between the different models discussed in this paper.

1.1 1D-Three-Component Examples

Two examples from Metcalfe and Yarborough (1979) are replicated using the IFT correction presented by Coats (1980). The three-component cases are one-dimensional with pure $\hbox {CO}_2$ injection at one end, and a producer at the opposite end. A controlled pressure setting is used for the wells with a pressure of 1 bar above and 1 bar below the initial condition for the injector and producer, respectively. The reason for this is to ensure that the phase envelope shown on the ternary diagram is representative throughout the simulation, and therefore captures the ‘true’ compositional path. Table 5 shows the conditions for each case. The grid used is made up of 1000 grid blocks ($0.1\times 10\times 10\,\hbox {m}$) with a permeability of 200 md, and a porosity of 20%. Very small time steps (0.01 days) are forced on the simulator to minimize the time truncation errors (max CFL number of 0.06). The simulation is run for a total of 80 days using the Peng–Robinson equation of state. The immiscible relative-permeability curves used are shown in Fig. 23; the 45$^\circ$ diagonals shown on the right of Fig. 1 are used as the miscible set.

Case #1 represents an immiscible displacement, since both the injection and initial conditions lie below the critical tie-line extension. We expect very small changes to the immiscible $k_r$ curves in this case. The compositional path is shown in Fig. 24, where each point represents a gridblock composition at the end of the simulation. The figure on the right shows the value of the interpolation parameter, $F_k$, for all the gridblocks in the compositional space with lower values near the critical point as expected. The slight difference in the simulation results is clearer for the $\hbox {CO}_2$ concentration front (Fig. 25). The corresponding correction parameter ($F_k$) at the end of the simulation is also shown with a minimum value around 0.7 indicating a more immiscible-like displacement.

Fig. 24

Ternary representation of the displacement in case 1 showing the compositional path (left) and the interpolation parameter $F_k$ values for the corrected case (right)

Fig. 25

$\hbox {CO}_2$ concentration distribution at the end of the simulation with and without a correction (left) and the corresponding interpolation parameter $F_k$ values (right)

Case #2 is a multi-contact miscible (MCM) displacement. The process involves a vaporizing gas drive mechanism since the initial composition is above the critical tie-line extension while the injection composition is below. We expect to see a larger impact of the correction in this MCM displacement compared to the immiscible displacement of Case #1. The compositional path enters the two-phase envelope near the critical point and follows the dew-point line down to the injection composition (Fig. 26). Figure 27 shows the $\hbox {CO}_2$ concentration front for both cases; the two plots overlay each other even though $F_k$ is 0.3 at the miscible front. The MCM displacements show smaller changes with the ‘correction’ than the immiscible case. The small changes in the results can be attributed to the very small two-phase region that is encountered in such displacements. Again, these cases shed light on the impact of not applying a correction in the super-critical region above the critical tie-line.

Fig. 26

Ternary representation of the displacement in case 2 showing the compositional path (left) and the interpolation parameter $F_k$ values for the corrected case (right)

Fig. 27

$\hbox {CO}_2$ concentration distribution at the end of the simulation with and without a correction (left) and the corresponding interpolation parameter $F_k$ values (right)

1.2 2D-4-Component Example

The 4-component system we consider, obtained from Orr et al. (1993), is made up of $\hbox {CO}_2-C_1-\hbox {n}C_4-\hbox {n}C_{10}$. The case is run on the upscaled layer 15 of the SPE10 model shown in Fig. 28. One injector well located in the top-left corner ($1\times 1$) is injecting pure $\hbox {CO}_2$ at a fixed BHP of 140 bars. The single producer is placed in the opposite corner of the grid ($30\times 110$) producing at a fixed BHP of 80 bars. The initial condition of the system is made up of 10% $C_1$, 20% $\hbox {n}C_4$ and 70% $\hbox {n}C_{10}$ at 110 bars and 344 K. The relative permeability curves used in this model is the same one shown in Fig. 16.

Fig. 28

$\hbox {Log}{(k_x)}$ map of upscaled 15th layer of SPE10—block size: 12.2 $\times 6.1 \times 1.2\,\hbox {m}$

Fig. 29

Gibbs free energy of gas (top-left) and oil (top-right) and the ratio of oil to gas (bottom) in the two phase region of the quaternary system at a pressure of 110 bars and 344 K

We use the same cases with the same parameters in Sect. 4.1 except for the reference Gibbs free energy that is 3.12 for the GIBBS-ALL case. A limitation in our implementation is our use of a single reference Gibbs free energy for the whole simulation run. This value should change with pressure and temperature, and in the 4-component case should capture the value of the closest critical point along the locus of critical points.

Figure 29 shows the gas and oil Gibbs free energy in the two-phase region showing the same trend observed in the three-component systems. The Gibbs free energy of the gas is always higher than the oil in the two-phase region and decreases as the critical locus is approached. The Gibbs free energy of the oil increases as the critical locus is approached. We also show the ratio of the two approaching one around the critical locus that confirms that the behavior seen in three-component systems is applicable to multi-component systems. The reference value of 3.12 used in this case is the maximum oil/minimum gas Gibbs free energy in the two-phase region.

The case was run for a total time of 7000 days with a maximum time step of 500 days, this resulted in a max average CFL number of 40 for the IFT and GIBBS cases, and 60 for the GIBBS-ALL case. The normal case in this example is unable to complete the simulation run due to time steps cutting to very small values. Figure 30 shows the cumulative newton iterations for each case with the normal case facing difficulties with convergence shortly after two-phase flow develops. A more in depth analysis of the nonlinear performance is required to fully understand the reasons for this difficulty. Any correction applied to the relative permeability in the near-miscible region significantly improves nonlinear convergence.

Fig. 30

Cumulative newton iterations for the 4-component example

Fig. 31

Gas saturation distribution at the end of the simulation after 7000 days for the different cases with the corresponding $F_k$ map

Fig. 32

Gas production rates (top) and quaternary representation of the compositional path for the GIBBS-ALL case showing $F_k$ values with a phase envelope constructed at a pressure of 110 bars and 344 K (bottom) with bubble points in red and dew points in blue

Figure 31 shows the gas saturation and corresponding $F_k$ maps. We see that not much correction is taking place in this example showing only a small region in the IFT case that has a surface tension less than 1 dyne/cm. Again, the GIBBS applies a very small correction on a larger area of the two-phase region with not much effect on the simulation results as evident in the gas production rates in Fig. 32. The GIBBS-ALL case applies some correction in the single-phase region that is close to the critical locus near the injection composition ($\hbox {CO}_2$) shown on the quaternary representation in Fig. 32. We observe the same behavior of the previous three-component cases; a significant improvement in the convergence rate when applying any correction with slight differences in gas production rates.