Global Riemann Solver and Front Tracking Approximation of Three-Component Gas Floods

We study a 2× 2 system of non-strictly hyperbolic conservation laws arising in three– component gas flooding for enhanced oil recovery. The system is not strictly hyperbolic. In fact, along a curve in the domain one family is linearly degenerate, and along two other curves the system is parabolic degenerate. We construct global solutions for the Riemann problem, utilizing the splitting property of thermo-dynamics from the hydro-dynamics. Front tracking simulations are presented, using the global Riemann Solver. 1 The Three-Component Gas Flooding Model We consider a simplified compositional displacement model for a three-component system at constant temperature and pressure [11], (1.1) (C1)t + (F1(C1, C2))x = 0 , (C2)t + (F2(C1, C2))x = 0 , associated with initial data (1.2) C1(0, x) = C̄1(x) , C2(0, x) = C̄2(x) . The independent variables (t, x) are normalized such that the overall velocity is 1. Here Ci is the overall ith component volume fraction, and Fi is the overall i th component flux. For the third component, we trivially have C3 = 1− C1 − C2 , F3 = 1− F1 − F2 . The couplet (C1, C2) takes values in a triangular domain D = {(C1, C2) | C1 ≥ 0, C2 ≥ 0, 1− C1 − C2 > 0} . For the phase behaviors that are considered in this paper, there exists a subset D2 ⊂ D, referred to as the two-phase region, where the fluid splits into two phases, the liquid and the gaseous phases. In the single phase region D1 = D \D2, we trivially have F1(C1, C2) = C1, F2(C1, C2) = C2.