Orr, F. M. - Theory of Gas Injection Processes Episode 5

4.8. EXERCISES 71 Binary Displacement with Mutual Solubility. The discussion of the application of the velocity constraint and entropy condition to eliminate nonphysical solutions follows that of Johns [54, Chapter 3] for the Buckley-Leverett problem. Examples of solutions for displacement of C10 by CO2 are given by Pande [95, Chapter 4]. The description of the dependence of solutions on initial and injection conditions was given first by Helfferich [32]. Effects of Volume Change on Mixing. A comparison of binary solutions with and without volume change as components change phase is given for CO2/C10 displacements by Dindoruk [19, Chapter 6]. 4.8 Exercises 1. Characteristic curves. Consider the equation ∂t + C2 ∂x = 0. (4.8.1) The initial composition is Cinit = 0.1, and the injection composition is Cinj = 0.8. Derive expressions for the characteristic curves. Plot the appropriate characteristic curves on a t-x diagram. Determine whether shocks would occur in this displacement. 2. Gas dissolution. Consider a laboratory core which contains initially water that is saturated with CO2 in equilibrium with gas at the critical gas saturation, Sgc = 0.05. The equilibrium volume fraction of CO2 dissolved in the saturated water phase is 0.03, and the volume fraction of water in the gas phase is 0.001. At time τ = 0, injection of pure water into the core begins. Assuming that effects of volume change as components change phase can be neglected, calculate the saturation profile at τ = 0.5 pore volumes injected. Determine how much pure water would have to be injected to remove all the gas present initially in the core. 3. Calculate the saturation profile at τ = 0.5 and 1.0 pore volumes injected for the relative permeability functions of Eqs. 4.1.14-4.1.19 with Sgc = 0.1, Sor = 0.3, and M = 10 for a displacement in which gas displaces oil. The volume fraction of the light component required to saturate the liquid phase is 0.4, and the volume fraction of light component in the equilib-rium vapor phase is 0.95. The initial composition is a single-phase mixture in which the light component has a volume fraction of 0.2. The volume fraction of the light component in the injection gas is 0.98. Also calculate a recovery curve for the heavy component. How many pore volumes of gas must be injected to recover all of the oil initially in place? 4. Displacement with two-phase initial and injection mixtures. Consider the fluid system of problem 3. Calculate the saturation profiles at the same times and calculate a recovery curve for a displacement in which the core initially contains a two-phase mixture in which the volume fraction of the light component is 0.5, and the injection gas is also a two-phase mixture with a light component volume fraction of 0.9. 5. Effect of volume change on shock speed. Consider the situation outlined in problem 2. De-termine the shock speed for a situation in which the density of the water does not change as it moves between phases, but CO2 that dissolves in the water phase occupies only half the 72 CHAPTER 4. TWO-COMPONENT GAS/OIL DISPLACEMENT Table 4.4: Equilibrium Phase Compositions and Fluid Properties Fluid Initial Oil Equil. Liq. Equil. Vap. Injected Gas xCH4 xC10 0. 1. 0.3519 0.6481 0.9964 0.0036 1. 0. ρ (gmol/l) 4.881 6.481 4.316 4.298 ρ µ (g/cm3) (cp) 0.6945 -0.6342 0.262 0.0712 0.015 0.0690 - volume of CO2 in the vapor phase. Assume that the CO2 in the vapor phase has a compo-nent density of 1.1364 x 10−3 gmol/cm3 and that water in the liquid phase has a component density of 5.5555 x 10−2 gmol/cm3. 6. CH4 displacing C10. Consider the fluid property data givenin Table 4.4 for the CH4/C10 system at 160 F and 1600 psia. Calculate and plot the composition profile as a function of ξ/τ for two differing assumptions about density behavior: (1) when the volume occupied by each component is the pure compo-nent volume no matter what phase the component appears in, and (2) when the equilibrium phases assume the densities given in Table 4.4. Assume that the phase compositions in mole fractions given in Table 4.4 are correct for both cases. Calculate and plot recovery curves as a function of pore volumes of methane injected for the two density assumptions. Chapter 5 Ternary Gas/Oil Displacements In this chapter, we consider the behavior of displacements in which three components and two phases are present. Much of the original work on gas drives was done for ternary systems, which display essential features of displacement behavior but are simple enough to analyze. The basic physical mechanisms of gas drives were outlined by Hutchinson and Braun [38], who considered what would happen if a porous medium were represented as a series of mixing cells. Figs. 5.1 and 5.2 summarize their arguments. (For another version of the mixing cell argument, see Lake [62].) Suppose that oil with composition O1 is displaced by gas with composition G1. Mixtures of oil O1 with gas G1 in the first mixing cell lie on the dilution line that connects O1 to G1 on the ternary diagram in Fig. 5.1. Suppose that after mixing some oil with gas in the first cell, the overall composition is M1. That mixture splits into two phases with compositions V1 and L1. Now assume that the less viscous vapor phase with composition V1 moves to the next downstream cell, where it mixes with fresh oil. Those mixtures lie on the dilution line that connects V1 with the oil composition, O1. If the new overall composition is M2, then the phases that form in the second cell have compositions L2 and V2. But when the vapor V2 moves to the next cell and mixes with fresh oil, the dilution line does not pass through the two-phase region. Instead, the mixtures are “miscible” after multiple contacts, even though the original gas and oil do not form only one phase when mixed in any proportions. This displacement is what is known as a vaporizing gas drive because the crucial transfer of components that leads to miscibility is the vaporization of the intermediate component from the oil into the fast-moving vapor phase. Mixture V1 is richer in component 2 than the original injection gas is, and mixture V2 is richer still. Oil O1 is rich enough in component 2 that miscibility develops. If, however, the oil had had composition L2 (or any mixture on the extension into the single-phase region of the tie line that connects V2 and L2), mixture of V2 with fresh oil would have given another mixture on the same tie line. In that case, the enrichment of the vapor phase with component 2 ceases to change with further contacts in downstream mixing cells. Such a vaporizing gas drive is said to be “immiscible.” In vaporizing gas drives, the mixing cell argument indicates that miscibility develops if the original oil composition does not lie within the region of tie line extensions on the ternary diagram. Fig. 5.2 summarizes a similar argument for a displacement known as a condensing gas drive in which gas G2 displaces oil with composition O2. Mixtures of original oil with gas in the first mixing cell give composition M1. That mixture splits into phases with compositions V1 and L1. Here again, the vapor phase is assumed to move ahead and contact fresh oil, but this time we focus 73 74 CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS C1 a 1 V1 1 V2 M2 a a L1 L2 C3 a O1 C2 Figure 5.1: Mixing cell representation of a vaporizing gas drive. on what happens in the first mixing cell. There, liquid phase with composition L1 mixes with new injected gas. Those mixtures lie on the dilution line that connects L1 with G2. If the new composition after mixing in the first cell is M2, then the resulting phase compositions are V2 and L2. When liquid phase L2 mixes with new injection gas, the mixtures are single-phase. Here again, multiple contacts of the gas with the oil have created mixtures that are miscible. Thisdisplacement is called a condensing gas drive because it relieson transfer of the intermediate component from the injected gas phase to the oil. Ternary condensing gas drives are multicontact miscible if the injection gas composition lies outside of the region of tie line extensions on a ternary phase diagram. If, on the other hand, the injection gas had had some composition on the extension of a tie line (say the L2-V2 tie line), the enrichment of the oil in the first cell with component 2 condensing from the gas would have stopped when that tie line was reached, because additional mixtures of fresh gas with liquid L2 would fall on the same tie line again. The mixing cell argument is necessarily a qualitative one because it assumes that only the vapor phase moves from cell to cell. In a displacement in a porous medium, the phases would move according to their fractional flows. Our task for this chapter, then, is to put the qualitative argument of Hutchinson and Braun on a firm mathematical footing. The Riemann problem we will consider is illustrated in Fig. 5.3: for a given pair of initial and injection compositions, find the set of compositions that form between the injection composition at the upstream end of the transition zone and the initial composition at the downstream end. When three components are present, the flow is no longer constrained to take place along a single tie line, as it is in binary displacements. Thus, for three-component flows, an essential part of the problem is to find the collection of tie lines (and their associated phase compositions) that are traversed during a displacement. Three-component flows have been considered for systems that range from alcohol displacements [125] to surfactant flooding [35] to gas/oil systems [134, 22]. The ideas developed in this and the next chapter come from many sources reviewed at the end of the chapter. The development given here draws heavily from the work of Johns [54, Chapter 3], Dindoruk [19, Chapters 3, 4, and 6], and Wang [128]. 5.1. COMPOSITION PATHS 75 C1 a 1 a 2 O2 a M1a M2 a a L1 L2 aG2 C3 C2 Figure 5.2: Mixing cell representation of a condensing gas drive. We begin by formulating the eigenvalue problem that determines wave velocities and allowed composition variations, and we develop the idea of a composition path. Next we consider the behavior of shocks, which play important roles in the behavior of solutions. In Sections 5.3 and 5.4, example solutions are described that show in detail the patterns of flow behavior associated with vaporizing and condensing gas drives. Section 5.5 shows that the key patterns of shocks and rarefactions (continuous composition variations) for ternary systems can be catalogued in a simple way based on the lengths of two key tie lines and whether tie lines intersect on the vapor side or the liquid side of the two-phase region. Section 5.6 introduces the important concept of multicontact miscibility. Effects on ternary systems of volume change as components transfer between phases and calculation of component recovery are reviewed in the remaining sections. 5.1 Composition Paths The conservation equations for a three-component system without volume change are ∂C1 ∂F1 ∂τ ∂ξ ∂C2 ∂F2 ∂τ ∂ξ where = 0, (5.1.1) = 0, (5.1.2) Ci = ci1S1 + ci2(1 −S1), i = 1,2, (5.1.3) and Fi = ci1f1 + ci2(1 −f1), i = 1,2. (5.1.4) ... - tailieumienphi.vn