Orr, F. M. - Theory of Gas Injection Processes Episode 8

5.11. EXERCISES 131 the properties of envelope curves to determine when self-sharpening behavior occurs. Dindoruk [19] and Wang [128] derived the closed form analytical solutions for constant K-values. Dindoruk [19] proved that tie lines connected by a shock must intersect when effects of volume change are included in the model. 5.11 Exercises 1. Calculate and plot the liquid locus and the vapor locus for a ternary system in which the K-values are constant at K1 = 3.0, K2 = 1.2, and K3 = 0.01. 2. Show that the liquid phase composition of a tie line that extends through a point (C∗, C∗) is given by x1 = −b ±√b2 −4ac, (5.11.1) where (K1 − K3) (K1 −K2) (K2 − K3) (K2 −1) (K1 − K3) ∗ (K1 −1) ∗ (1 − K3) (K1 − K2) (K2 − K3) (K2 −1) (K2 −K3) (K2 − 1) and (1 − K3) (K2 −K3) 1 3. Problems 3-12 construct the solution for a HVI vaporizing gas drive. Find the tie lines that extend through the injection composition, Cinj = 0.9, Cinj = 0.1 and the initial composition Cinit = 0.1, Cinit = 0.2 for a ternary system with the K-values of Problem 1. 4. For a viscosity ratio, µliq/µvap = 10, and the relative permeability functions of 4.1.14-4.1.19 with Sor = Sgc = 0, calculate and plot the tie-line and nontie-line eigenvalues as a function of saturation for a tie line that extends through the point Cinit = 0.1, Cinit = 0.2 with K1 = 3.0, K2 = 1.2, and K3 = 0.01. 5. Calculate the compositions and velocity of a leading shock from the initial composition of Problem 3 into the two-phase region for the K-values, relative permeabilities, and viscosities of Problem 4. 6. Find the composition on the tie line of Problem 5 at which the tie-line and nontie-line eigen-values are equal. Calculate the compositions and velocities of the rarefaction along the initial tie line. 132 CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS 7. Evaluate and plot the compositions on the nontie-line path that connects the initial and injection tie lines of Problem 3. Find the composition at which the nontie-line path intersects the injection tie line. 8. Determine the compositions and velocity of a trailing semishock for the injection tie line (with the data of Problem 4). 9. Determine whether a rarefaction occurs along the injection tie line. If so, calculate the compositions and wave velocities for that rarefaction. If not, calculate the velocity of the trailing shock. 10. Plot a ternary diagram that shows the composition route determined in Problems 3-9. 11. Plot a saturation profile for the solution of Problems 3-9. Also plot composition profiles for each of the components. 12. Plot the recovery of components 2 and 3 as a function of pore volumes injected. 13. Problems 13-20 construct the solution for a HVI condensing gas drive. Find the tie lines that extend through the injection composition, Cinj = 0.7, Cinj = 0.3 and the initial composition Cinit = 0.1, Cinit = 0.2 for a ternary system with the K-values of Problem 1. 14. Calculate the compositions and velocity of a trailing shock from the injection composition of Problem 13 into the two-phase region. Use the K-values of Problem 1 and the relative permeabilities and viscosity ratio of Problem 4. 15. Find the composition on the upstream side of the intermediate shock and evaluate the com-positions and saturation for the rarefaction on the injection tie line. 16. Find the composition on the downstream side of the intermediate shock. Determine whether a rarefaction exists on the initial tie line, and if so, evaluate the compositions and saturations that make up the rarefaction. 17. Determine the composition on the upstream side of the leading shock, and find the shock velocity. 18. Plot a ternary diagram that shows the composition route determined in Problems 13-17 19. Plot saturation and composition profiles for each component for the solution obtain in Prob-lems 13-17. 20. Plot recovery curves for each of the components for the solution of Problems 13-17. 21. LVI vaporizing gas drive. Calculate and plot composition path and saturation and composi- tion profiles and recovery curves for a constant K-value system with K = 3.0, K = 0.9, and K3 = 0.01. The injection composition is Cinj = 0.9, Cinj = 0.1 and the initial composition is Cinit = 0.1, Cinit = 0.2. Use the relative permeabilities and viscosity ratio of Problem 4. 5.11. EXERCISES 133 22. LVI condensing gas drive. Calculate and plot composition path and saturation and composi- tion profiles and recovery curves for a constant K-value system with K = 3.0, K = 0.9, and K3 = 0.01. The injection composition is Cinj = 0.7, Cinj = 0.3 and the initial composition is Cinit = 0.1, Cinit = 0.2. Use the relative permeabilities and viscosity ratio of Problem 4. 23. Show that the solution for a nontie-line path in a ternary system with constant K-values and Sgc = 0 and Sor = 0 is given by x1 = x1f1 − S1 + γ(f1 −S1 − 1) + γ(K−−1)(S1f1 − S1f1) +2γ(K1 − 1) Z S1 f1dS1, (5.11.2) 1 1 1 with ZSS1 f1dS1 = S1 −S0 1 Smax −1 ⎧ ⎛(S −S ) 1 + 1− Smax ⎞ + arctan q 1 1 max M M ⎛(S0 −S ) 1 + 1 − Smax ⎞⎫ −arctan q max M Smax ⎨ M (Smax)2 − M (Smax)(S1 −Sgc) + 1 + M (S1 − Sgc)2 ⎬ M + 1 2 ⎩ 1 (Smax)2 − 2 (Smax)(S0 −Sgc) + 1 + 1 (S0 −Sgc)2⎭ (5.11.3) where Smax = 1 − Sor −Sgc. 24. Find and plot the composition path and the saturation and composition profiles for a system containing CH4, C4, and C10 at 160 F and 1600 psia, including the effects of volume change as components change phase. The injection gas is a binary mixture of CH4 and C4 containing 90 mol percent CH4, and the initial mixture has composition C1 = 0.1, C2 = 0.4, C3 = 0.5. 134 CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS 25. Consider a displacement in a ternary system in which volume change is not important. The fractional flow functions for the initial and injection tie lines have the form shown in Fig. 5.19. Assume that the initial composition is fixed in the liquid portion of the single-phase region, but the injection gas composition can take any value along the injection tie line. Consider injection compositions that lie inside the two-phase region as well as in the single-phase regions on the liquid and vapor sides of the two-phase region. Sketch the solution composition paths and profiles for each of the possible patterns of displacement behavior and identify the transition points between them. Explain why the transition points mark the change in displacement pattern. 26. Show that the equal eigenvalue pints, λt = λnt observed as the saturation varies along a fixed tie line occur at a miximum or a minimum in λnt. 27. Show that for a fixed tie line in an LVI system, the following statements hold: C1 < C1 < C1 , λnt > 1, C1 < C1 < C1 , λnt < 1. In these expressions, C1 is the volume fraction of component 1 on the liquid locus, C1 is the composition of the intersection of the equivelocity curve with the tie line, and C1 is the composition on the vapor locus. 28. Show that for a fixed tie line in an HVI system, the following statements hold: C1 < C1 < C1 , λnt < 1, C1 < C1 < C1 , λnt > 1. Chapter 6 Four-Component Displacements The development of the theory for four-component displacements follows directly from the ideas presented in Chapters 4 and 5 but with some important additions. As in the simpler two- and three-component systems, we will formulate an eigenvalue problem. Here again, the eigenvalues represent propagation velocities for a given overall composition, and eigenvectors are allowable directions of composition variation in three-dimensional composition space. When four components are present, however, there are three eigenvalues and three eigenvectors for each point in the composition space. Thus, the problem is to find the unique composition path that connects the injection gas and initial oil compositions in a three-dimensional composition space. Much of the machinery needed to solve the four-component problem is already in place, so the development here will focus on the new features that arise in the more complex displacements. The most important difference between ternary and quaternary displacements is that a new key tie line appears. The initial oil and injection gas tie lines are still important, but parts of the solution behavior depend on a third tie line known as the crossover tie line. We begin by considering the eigenvalue problem and the composition paths that result. Here again, the special case of constant equilibrium K-values will prove useful. Next we use four-component solutions to understand dis-placement behavior that has come to be know as condensing/vaporizing drives (Zick [140]), and we consider how miscibility develops in four-component systems. Finally, we consider again the effects of volume change on mixing. 6.1 Eigenvalues, Eigenvectors, and Composition Paths 6.1.1 The Eigenvalue Problem When four components are present, there are three independent equations (when effects of volume change as components change phase are not considered, four independent equations if volume change is included): ∂C1 ∂F1 ∂τ ∂ξ ∂C2 ∂F2 ∂τ ∂ξ = 0, (6.1.1) = 0, (6.1.2) 135 ... - tailieumienphi.vn