Analytical solutions for gas displacements with bifurcating phase behavior

Minimum miscibility pressure (MMP) is one of the most important parameters in the design of a successful gas flooding process. The most reliable methods to calculate the MMP are based on slimtube experiments, 1-D slim-tube simulations, mixing cell calculations, and the analytical methods known as the method of characteristics (MOC). The calculation of MMP using MOC is the fastest method because it relies solely on finding the key tie lines in the displacement path. The MOC method for MMP estimation in its current form assumes that the composition path is a series of shocks from one key tie line to the next. For some oils, however, these key tie lines do not control miscibility and the MMP calculated using the key tie line approach can be significantly in error. The error can be as high as 5,000 psia for heavier oils or CO₂ displacements at low temperature where three-phase hydrocarbon regions can exist (L₁-L₂-V). At higher pressures, the two- or three-phase region can split (or bifurcate) into two separate two-phase regions (L ₁-L₂ and L₁-V regions). Thus, for the MMP calculation from MOC to be correct we must calculate the entire composition path for this complex phase behavior, instead of relying on the shock assumption from one key tie line to the next. In this paper the MOC composition route is developed completely for the bifurcating phase behavior displacement using a simplified pseudoternary system that is analogous to the complex phase behavior observed for several real displacements with CO₂. We develop the MOC analytical solutions by honoring all constraints required for a unique solution; velocity, mass balance, entropy, and solution continuity. The results show that a combination of shocks and rarefaction waves exist along the nontie-line path, unlike previous MOC solutions reported to date. We show that by considering the entire composition path, not just the key tie lines, the calculated MMP agrees with the mixing cell method. We also show that in this complex ternary displacement the displacement mechanism has features of both a condensing and vaporizing drive, which was thought to be possible only for gas floods with four or more components. For pure CO₂ injection the solution also becomes discontinuous for oils that lie on the tie-line envelope curve. Finally, we show that shock paths within the two-phase region are generally curved in composition space and that there is no MMP for some oil compositions considered in the displacements by CO₂. Recovery can be large even though the MMP is not reached.