University:
Boston University
Course:
ENG ME 303 | Fluid Mechanics
Academic year:

2023
Views:

290

Pages:

8
Author:

Ryder Coleman

Math, Num and Comp Models Black–Oil Solution Algorithm References Math, Num and Comp Models Compositional Oil Table of contents 1 Math, Num and Comp Models Processes to be modeled Reservoir Properties One–Phase Flow Two–Phase Immiscible Flow Black–Oil Compositional Oil Solution Schemes 2 References References Math, Num and Comp Models References Compositional Oil Assumptions: Compositional flow involves many components and mass transfer between phases in a general fashion. The flow process is isothermal. The components form at most three phases. There is no mass transfer between the water phase and the hydrocarbon phases. Total mass is conserved for each component: Water component: (φξw Sw ) + ∇ · (ξw ~uw ) = qw ∂t where ξw is the molar density of the water. Math, Num and Comp Models References Compositional Oil Hydrocarbon components: φ(xio ξo So + xig ξg Sg ) + ∇ · (xio ξo ~uo + xig ξg ~ug )+ ∂t ∇ · (dio + dig ) = qi , i = 1, . . . , Nc where ξio , xiα and diα represents the molar densities, the mole fraction, and the difussive fluxes of componet i in the phase α, respectively. α = o, g. Nc is the number of hydrocarbon components, Darcy’s Law: ~uα = − krα k(∇pα − ρα G∇D) for α = w, o, g. µα Math, Num and Comp Models References Compositional Oil The mole fraction balance implies: Nc X xio = 1 and i=1 Nc X xig = 1 i=1 Saturation constrain: Sw + So + Sg = 1. Capillary pressures: pcow = po − pw ; pcgo = pg − po . We have Nc + 9 equations and 2Nc + 9 unknowns: xio , xig , ~uα , ρα and Sα , for α = w, o, g and i = 1, . . . , Nc . The aditional Nc relations are provided by the equilibrium relations that relate the numbers of moles: fio (po , x1o , . . . xNc o ) = fig (pg , x1g , . . . xNc g ) Typical (moderate) simulation: grid nodes ∼ 105 and Nc = 10 =⇒∼ 3 × 106 unknowns by time step. Math, Num and Comp Models Solution Schemes Table of contents 1 Math, Num and Comp Models Processes to be modeled Reservoir Properties One–Phase Flow Two–Phase Immiscible Flow Black–Oil Compositional Oil Solution Schemes 2 References References Math, Num and Comp Models References Solution Schemes Solution Schemes I An important problem in the numerical simulation is to develop stable, efficient, robust, accurate, and self-adaptive time stepping techniques. IMPES method. This scheme works well for problems of intermediate difficulty and nonlinearity (e.g., for two-phase incompressible flow) and is still widely used in the petroleum industry. However, it is not efficient for problems with strong nonlinearities, particularly for problems involving more than two fluid phases. Simultaneous solution (SS) method. Solves all of the coupled nonlinear equations simultaneously and implicitly. This technique is stable and can take very large time steps while stability is maintained. Math, Num and Comp Models References Solution Schemes Solution Schemes II For the black oil and thermal models (with a few components) is a good choice. However, for complex problems that involve many chemical components (e.g., the compositional and chemical compositional flow problems), the size of system matrices to be solved is too large. Sequential methods, implicit fashion without a full coupling. They are less stable but more computationally efficient than the SS scheme, and more stable but less efficient than the IMPES scheme. The sequential schemes are very suitable for the compositional and chemical compositional flow problems that involve many chemical components. Adaptive implicit scheme can be employed in reservoir simulation.