Implementation of a Streamline Method For Flow Simulation of Large Fields
Authors: Behrens R.A. , Jones R.C. , Emanuel A.S. 
Published: March 2010

Abstract
This paper discusses the implementation of an efficient streamline method that is regularly employed where minimal modelling effort is desired or where the problem is not practical for traditional finite-difference simulation. Two streamline methods are described with full details given for one method. The method detailed uses a finite-difference 2D stream function, open or closed boundaries, fractured or unfractured wells, streamline tracking by contouring, volumetric analysis by geometry, and single-phase resistance by geometry. These steps are contrasted with an alternative method. Results from both streamline methods are shown for two cases:a five-spot for comparison with an analytical solution anda skewed pattern element to illustrate a minimal modelling application.
Results of one method are also given for the Windalia sand waterflood. The Windalia sand is a tight heterogeneous sandstone developed on 4 to 8 ha (20 to 40 acre) spacing with 600+ wells. The 30 year waterflood history is characterized by realignment from ninespot to line drive patterns, hydraulically fractured producers, infill drilling, and stress induced fracturing at the injectors. The Windalia sand case illustrates a problem too complex for traditional finite-difference simulation.

Categorization of Streamline Methods
Two categories of streamline methods are summarized in Table 1. Method A and Method B can each be separated into six steps. This categorization both gives an overview of the options and serves as a common vocabulary. Listing variations within each broad category listed here could split this table much finer. The steps of Method A are described in detail in this paper and contrasted briefly with the corresponding steps of Method B.
The six steps are sequential with each step adding a new capability to the analysis: Step 1 forms the underlying stream function field, Step 2 allows a visual picture of the pathlines, Step 3 allows for single-phase displacement calculations, and Step 4 forms the basis for multiphase displacement calculations done in Step 5. Step 6 extends the fixed 1D streamlines to change with time by an approximate saturation remapping.

Review of Previous Work
Streamlines and streamtubes fall within potentiometric methods that were described in the early monograph of Muskat(1). Streamtubes were first introduced by Higgins and Leighton(2) in 1962 for waterflooding using Method A of Table 1. They divided a five-spot along streamlines and mapped a Buckley-Leverett function(3) onto the streamlines treating them as 1D conduits. LeBlanc and Caudle(4) alternatively used Method B of Table 1 preferring to calculate pressure to stream function. Martin and Wegner(5) used Method A with a finite-difference stream function. Other semi-analytical techniques have been introduced to generate the pressure field for streamline tracking in "sectionally homogenous"(6) permeability fields. One of these(7) was presented with the rationale that "methods based on finite-difference techniques are difficult to implement, because they require substantial post-processing for front tracking and cannot handle singularities near the well." However, as shown in Table 1, the initial calculation of pressure or stream function is independent of the post-processing which occurs in Steps 2 through 5.