Numerical Reservoir Simulation Using an Ordinary-Differential-Equations Integrator
Authors: Sincovec R.F. 
Published: December 2007

Abstract
The method of lines used in conjunction with a sophisticated ordinary-differential-equations integrator is an effective approach for solving nonlinear, partial differential equations and is applicable to the equations describing fluid flow through porous media. Given the initial values, the integrator is self-starting. Subsequently, it automatically and reliably selects the time step and order, solves the nonlinear equations (checking for convergence, etc.), and maintains a user-specified time-integration accuracy, while attempting to complete the problems in a minimal amount of computer time. The advantages of this approach, such as stability, accuracy, reliability, and flexibility, are discussed. The method is applied to reservoir simulation, including high-rate and gas-percolation problems, and appears to be readily applicable to problems, and appears to be readily applicable to compositional models.

Introduction
The numerical solution of nonlinear, partial differential equations is usually a complicated and lengthy problem-dependent process. Generally, the solution of slightly different types of partial differential equations requires an entirely different computer program. This situation for partial differential equations is in direct contrast to that for ordinary differential equations. Recently, sophisticated and highly reliable computer programs for automatically solving complicated systems of ordinary differential equations have become available. These computer programs feature variable-order methods and automatic time-step and error control, and are capable of solving broad classes of ordinary differential equations. This paper discusses how these sophisticated ordinary-differential-equation integrators may be used to solve systems of nonlinear partial differential equations. partial differential equations.The basis for the technique is the method of lines. Given a system of time-dependent partial differential equations, the spatial variable(s) are discretized in some manner. This procedure yields an approximating system of ordinary differential equations that can be numerically integrated with one of the recently developed, robust ordinary-differential-equation integrators to obtain numerical approximations to the solution of the original partial differential equations. This approach is not new, but the advent of robust ordinary-differential-equation integrators has made the numerical method of lines a practical and efficient method of solving many difficult systems of partial differential equations. The approach can be viewed as a variable order in time, fixed order in space technique. Certain aspects of this approach are discussed and advantages over more conventional methods are indicated. Use of ordinary-differential-equation integrators for simplifying the heretofore rather complicated procedures for accurate numerical integration of systems of nonlinear, partial differential equations is described. This approach is capable of eliminating much of the duplicate programming effort usually associated with changing equations, boundary conditions, or discretization techniques. The approach can be used for reservoir simulation, and it appears that a compositional reservoir simulator can be developed with relative ease using this approach. In particular, it should be possible to add components to or delete components possible to add components to or delete components from the compositional code with only minor modifications.
SPEJ
P. 255