Multiphase Streamline Modeling in Three Dimensions: Further Generalizations and a Field Application
Authors: Peddibhotla Sriram , Datta-Gupta Akhil , Xue Guoping 
Published: December 2007

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Abstract
Recently three dimensional streamline modeling of multiphase flow has gained increasing popularity. It has also become an important tool as a fast and reliable flow simulator for solving inverse problems in reservoir characterization. However, field scale application of three dimensional streamline modeling has been rather limited. The difficulties with regard to field applications have been changing well configurations as a result of infill drilling, zone isolations, recompletions, etc. A critical issue here is the remapping of streamlines and hence, fluid saturations as the dynamics of field conditions dictate. Past efforts to handle infill drilling during streamline simulation have been to average streamlines over an underlying grid and then to proceed with numerical computations along streamlines. Such an approach undermines one of the major strengths of streamline modeling, which in addition to the fast solution, is to preserve the self-sharpening nature of the saturation fronts during waterflooding Averaging of streamlines in conjunction with the lower order numerical solution of multiphase flow equations along streamlines lead to a significant loss in accuracy. We present two major improvements to the existing streamline modeling. First, instead of averaging streamlines during changing well conditions, we used a 3-D mapping algorithm for streamlines. Thus, we perform a mapping of streamlines to streamlines rather than streamlines to grid blocks whenever field conditions change. Second, along streamlines we use a third-order Total Variation Diminishing (TVD) scheme to solve the multiphase flow equations to minimize numerical dispersion, and prevent any nonphysical oscillations. We demonstrate these improvements by comparison with high resolution numerical simulations using both synthetic as well as field examples. The results clearly indicate the power and versatility of the approach for large-scale field application.

Introduction
Streamline approach to modeling multidimensional, multiphase flow essentially comprises of two steps: generating streamlines in 3D space and then solving the 1D governing equations analytically or numerically along the streamlines. Streamlines can be generated from an underlying velocity (and thus, pressure) field using the transit time algorithm as outlined by Datta-Gupta and King. Multiphase flow equations can then be solved in travel time coordinates which greatly facilitate analytical as well as numerical solution. These aspects of streamline modeling are further explored in this paper.
The streamline approach applied to waterflooding essentially decouples the pressure and saturation solutions. Since for typical waterfloods the total mobility is a rather weak function of saturation, the streamlines do not shift significantly with time and thus needs to be updated only infrequently. This can result in very significant savings in computation time as reported by several authors. Streamline simulation preserves sharp fronts by minimizing numerical dispersion and allows for dynamic data integration during reservoir characterization, as well as uncertainty evaluation using multiple geostatistical realizations. Another advantage is the simplicity with which streamlines may be used to visualize three-dimensional flow from arbitrary well patterns in heterogeneous reservoirs.
Till now, application of 3D streamline modeling of multiphase flow has been limited to simple pattern configurations, e.g. 5-spots or line drives. We report in this paper what we believe is the first 3D field application involving multiple patterns with irregular pattern geometry and changing well configurations. In addition, we present two major modifications made to an existing multiphase, three dimensional streamline simulator making it suitable for field scale waterflood applications. First, to prevent smearing of saturation and concentration fronts, we map these quantities from streamlines onto streamlines using a 3D interpolation algorithm that uses a Modified Quadratic Shepard method. The 3-D mapping algorithm is robust, computationally efficient and does not lead to any significant smearing of saturation fronts. Second, we implemented along each streamline a higher order numerical solution that is Total Variation Diminishing (TVD) and thus does not lead to non-physical oscillations while preserving the sharp fronts. This is achieved by discretizing each streamline along travel time coordinates and defining a third order flux term that includes anti-dispersion corrective term. P. 265^