## 1. Introduction

[2] Streamline based methods are widely used for various subsurface flow modeling problems, especially for advection-dominated displacements. To apply these methods it is necessary to construct streamlines and calculate the time of flight (TOF) along each streamline in an accurate and robust manner. This is usually achieved in three steps. First, a simulator or a numerical method is used to obtain the cell boundary face fluxes. Second, a suitable velocity field is interpolated throughout the entire computational domain. Third, streamlines are integrated using the velocity field, and the time of flight calculated along those trajectories. Of these three steps, the third step is the most straight-forward and utilizes either analytic solutions or Runge-Kutta techniques for the trajectory calculation. The first step is outside the scope of the current study. However, we will discuss enough of the flux calculation to lay a solid foundation for the discussion of the velocity interpolation models, which is the main subject of this paper.

[3] Unstructured grids are an important topic in reservoir simulation and three-dimensional (3-D) geologic modeling due to their flexibility especially for representing highly complex geologic structures and fluid flow near multilateral and fractured wellbore trajectories. In recent years, this research topic has received even more attention in the context of the emerging next generation reservoir simulators as well as the new data exchange standard for reservoir characterization, earth and reservoir models [*King et al.*, 2012]. It is both theoretically important and practically necessary to develop velocity interpolation models which are applicable in unstructured grids. Some of the frequently encountered unstructured grids in field applications are shown in Figure 1 and will be discussed in detail later.

[4] This paper is organized as follows. We begin with a literature review of the numerical calculation of flux and of velocity models. The next section is dedicated to a brief description of the problem to be solved, emphasizing the cell boundary conditions which then constrain the velocity interpolation discussions. The fourth section begins with an investigation of various velocity interpolation spaces in the unit square, and then discusses possible extensions of those velocity spaces to polygons. In the fifth section, several alternative methods based on subcell refinement are proposed. The rest of the paper evaluates the characteristics of the different velocity interpolation schemes and then demonstrates their application to several 2-D and 3-D reservoir modeling situations, including 2.5D PEBI grids, faulted grids, and grids with local refinement.