Computational methods for multiphase flows in porous media

Simulating multiphase flow in petroleum reservoir is a challenging task for applied and computational mathematicians. One needs to understand the fundamental partial differential equations describing the flow and transport of chemical species, the complex underlying geological structure of the reservoir, the appropriate numerical discretizations in space and time, and the efficient solution techniques with preconditioning for solving the resulting linear systems. In about 500 pages, this book succeeds in presenting and covering all of these topics. The authors are experts in this research field. The book is self-contained. However, because of the complexity of the task, it is understandable that the mathematical treatment is at the introductory level and theoretical proofs are not included. The plan of the book is as follows. Chapter 1, Introduction, briefly presents the main ideas covered in the book. Chapter 2, Flow and Transport Equations, gives a general description of several models: single phase flow, two-phase immiscible flow, transport of one or several components, black-oil model, oil volability, compositional flow, nonisothermal flow and flows in fractured porous media. Chapter 3, Rock and Fluid Properties, contains a nice summary of the different properties of the underlying reservoir. The rock properties include the capillary pressures, the relative permeabilities for two-phase and three-phase flows and the rock compressibility. The fluid properties include the formation volume factors, densities, solubilities, viscosities, equations of state (Peng-Robinson, Redlich-Kwong, RedlichKwong-Soave). Chapter 4, Numerical Methods, serves as a compact and useful introduction to the numerical methods that are popular in reservoir modeling. Important concepts and results are given without theoretical proofs. Emphasis is made on finite difference methods, standard continuous finite element methods, control volume finite element methods and mixed finite element methods. The chapter also covers briefly the discontinuous finite element methods and the characteristic finite element methods and it is ended by a discussion of issues relevant to adaptive mesh refinement. Chapter 5, Solution of Linear Systems, deals with several techniques for solving linear systems. In particular, the Krylov-based algorithms such as conjugate gradient, GMRES, orthomin, BiCGSTAB are presented. However, for realistic flow and transport simulations, the resulting matrix is often ill-conditioned and there is a need for preconditioning. One of the attractive parts in this chapter is the treatment of effective preconditioners. The rest of the book covers in detail the models introduced in Chapter 2. Chapter 6, Single Phase Flow, considers the flow of a slightly compressible fluid in the primary recovery process where oil is simply produced by natural decompression. First, a one-dimensional radial flow for which an analytical solution is obtained. Next, the nonlinear equation can be linearized by time-lagging the coefficients and it can be solved by the standard finite element method combined with a backward Euler time discretization. Another scheme is obtained by solving the nonlinear problem with a Newton-Raphson loop. Finally, using the forward Euler technique, an explicit scheme can be derived. The latter method is computationally cheaper but it requires a stability