# Mathematical Models and Finite Elements for Reservoir Simulation

## Single Phase, Multiphase and Multicomponent Flows through Porous Media

Numerical simulators for oil reservoirs have been developed over the last twenty years and are now widely used by oil companies. The research, however, has taken place largely within the industry itself, and has remained somewhat inaccessible to the scientific community. This book hopes to remedy the situation by means of its synthesized presentation of the models used in reservoir simulation, in a form understandable to both mathematicians and engineers.The book aims to initiate a rigorous mathematical study of the immiscible flow models, partly by using the novel `global pressure' approach in treating incompressible two-phase problems. A finite element approximation technique based on the global pressure variational model is presented, and new approaches to the modelling of various kinds of multiphase flow through porous media are introduced.Much of the material is highly original, and has not been presented elsewhere. The mathematical and numerical models should be of great interest to applied mathematicians, and to engineers seeking an alternative approach to reservoir modelling.

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Published: January 1986

Imprint: North-holland

ISBN: 978-0-444-70099-5

## Contents

• Basic Laws and Models for Flow in Porous Media. Generalities. The Geometry of the Field. The Basic Laws for One and Two-Phase Flow. The Basic Models. Qualitative Behavior of the Solution in the No-Diffusion and No-Capillary Pressure Case. Slightly Compressible Monophasic Fields. Construction of the Pressure Equation. Existence and Uniqueness Theorems. An Alternative Model of Monophasic Wells. Incompressible Two Phase Reservoirs. Construction of the State Equations. Summary of Equations of Two Phase Flows for Incompressible Fluids and Rock. An Alternative Model for Diphasic Wells. Mathematical Study of the Incompressible Two-Phase Flow Problems. The Case of Fields with Different Rock Types. Generalization to Compressible, Three Phase, Black Oil or Compositional Models. The Two-Phase Compressible Model. A Finite Element Method for Incompressible Two-Phase Flow. Approximation of the Pressure-Velocity Equations. Resolution of the Algebraic System for Pressure-Velocity. Approximation of the One-Dimensional Saturation Equation: The Case with Neither Capillary Pressure Nor Gravity. Approximation of the One-Dimensional Saturation Equation in the General Case. Approximation of the Saturation Equation in Two Dimensions.