This paper presents a new and fast approach for solving the tracer flow equations in oil reservoirs. It is based on the Fundamental Solutions Method (FSM) for the pressure equation and the exact solution of the concentration equations along the streamlines. The FSM combines a modified version of the Image's Method (MOI) and the Singular Value Decomposition algorithm (SVD) to obtain linear semianalytical estimates for pressure and velocity in the reservoir. These expressions are used to compute the streamline distribution and the time of flight for each streamline. The tracer concentration, in this work, is a multidimensional convection equation. It is transformed, by streamline-time of flight coordinate system, into a linear scalar one dimensional convection equation along each streamline. These equations are solved sequentially by the method of the characteristic to obtain the tracer concentration in the reservoir. The new approach was implemented and applied to study tracer flow problems in a synthetic heterogeneous reservoir. These studies showed that its main advantages are: no numerical diffusion in the approximation of the concentration, no need of stability condition on the time steps, and the reduction of CPU times in comparison with the finite and boundary element methods in reservoirs with many wells. In particular the new approach was found to be always faster than the boundary element method.
A new generation of streamline reservoir simulators based on an extremely efficient numerical solution of the tracer flow equations has emerged(1, 2). Its most intensive computational component is a finite differences or finite element multigrid solver used to determine the reservoir pressure. Some authors(3) consider this approach the mainstream technique to develop streamline reservoir simulations. In this article we present an original strategy for solving the tracer flow equations based on the FSM, which seems to be faster than mainstream techniques in some reservoir problems. FSM is not a standard numerical technique and there are several versions of it in the petroleum engineer literature(4, 5) without any reference to its real name. The best version, based on SVD, has been published by Marathe and coworkers(5) for homogeneous reservoirs. Its application to tracer flow equations is reported by Marathe(6) when the reservoirs heterogeneities are fractures and horizontal wells.
Recently, Marathe version of the FSM was extended to sectionally homogeneous reservoirs.(7, 8) An original application of this extension, to compute pressure and velocity, combined with the streamline time of flight(9, 10) (TOF) concept for solving convective tracer flow equations is explained and implemented in this paper. To our knowledged this approach has not been reported previously in the technical literature and it is different from Marathe streamtube scheme(6). Our work shows that in sectional homogeneous reservoirs with few regions and many wells this approach is the best. Therefore it represents a real alternative to solve reservoir engineering problems.
The paper is distributed in five sections. First section presents the convective tracer flow equations. A short review of the FMS is given in the second section. This is followed by an exposition of the streamlines approach in the third section. Finally, the numerical results and conclusion are given in the fourth and fifth sections.