document

Abstract

Traditional methods for streamline tracing are mainly based on the Pollock Cartesian scheme and its extensions, which require dual grids to treat general tensor fields on unstructured meshes. The complexity and inflexibility of dual grids impose substantial limitations on the application of streamlines. 

In this paper, we propose a computationally efficient method to construct streamlines on the original grid, taking flow fields from arbitrary schemes including Galerkin finite element methods. An essential component of the proposed streamline construction is fast processing of non-conservative velocity fields to recover or maintain local mass conservation and normal flux continuity simultaneously. Our construction involves balancing conservation residuals on each pair of adjacent elements using a Gauss-Seidel type iteration. Locally conservative velocities are extended from element faces to interiors using a local single element mixed finite element method.  Streamline and travel time are then obtained using standard methods of integration based on the Pollock algorithm. 

The proposed approach is shown to possess several advantages: it treats general tensor fields on unstructured and even nonmatching grids, maintains the optimal order of accuracy for high order elements, and avoids the difficulties imposed by nonphysical sinks and sources due to numerical inaccuracy. Several computational examples on structured and unstructured meshes are presented to demonstrate the effectiveness of the proposed method.

Introduction

Streamline-based methods have gained great popularity in reservoir simulations due mainly to their ability to treat large time steps, their small numerical diffusion effects, and their ability to incorporate semi-analytical one-dimensional solutions [e.g. Peddibhotla, Datta-Gupta and G. Xue, 1997; Batycky, Blunt and Thiele, 1996]. Consequently, they are able to speed the simulations of coupled flow and transport problems up to at least tenfold.  They are widely used for various subsurface modeling problems, in particular for modeling advection-dominated displacements. The accuracy and efficiency of streamline construction and travel time calculation are critical to the success of streamline-based simulations. Traditional methods for streamline tracing are mainly based on the Pollock Cartesian scheme and its extensions.  Although generalizations of the Cartesian tracing algorithm may treat general tensor fields on unstructured grids, a dual grid is usually required for utilizing the local conservation principle of finite volume methods.  Computation on a dual grid reduces the flexibility of the methods and increases the complexity of their implementation.

A computationally efficient method is proposed in this paper to construct streamlines on the original grid. It allows flow fields computed from arbitrary schemes including Galerkin finite element methods. The proposed algorithm is built upon the compatibility theory of algorithms for flow and transport [Dawson, Sun and Wheeler, 2004] and the compatibility-recovery projections of velocity [Sun and Wheeler, 2005a]. The velocity data, obtained from a flow scheme or by other means, are first processed to recover (or maintain) the local conservation and higher order compatibility conditions, while simultaneously recovering (or maintaining) normal flux continuity. We then trace the streamlines and compute the travel time using standard methods of integration based on the Cordes-Kinzelbach extension of the Pollock algorithm [Cordes and Kinzelbach, 1994]. 

The remaining parts of this paper are organized as follows. The flow equations considered are given in Section 2. Although we consider only single phase flow for simplicity of presentation, the algorithm presented here extends directly to multiphase flow. A few important and desired properties needed for a high quality streamline construction are discussed in Section 3.  The proposed streamline tracing algorithm, including the velocity processing, is presented in Section 4. The effectiveness of the new streamline tracing algorithm is illustrated using three numerical examples on structured and unstructured meshes. The importance of local mass conservation and normal flux continuity is also demonstrated by comparing the proposed algorithm with algorithms that do not satisfy these properties.