The implementation is given for an efficient streamline method that is regularly employed where minimal modeling 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 junction, 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 2 cases:
a five-spot for comparison with an analytical solution and
a skewed pattern element to illustrate a minimal modeling application. Results of one method are also given for case
the Windalia sand waterflood.
The Windalia sand is a tight heterogeneous sandstone developed on 20 to 40 acre spacing with 600+ wells. The 30 year waterflood history is characterized by realignment from 9spot 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.
Two categories of streamline methods are summarized in Table 1. Method A and Method B can each be separated into five steps. This categorization both gives an overview of the options and serves as a common vocabulary. This table could be split much finer by listing variations within each broad category listed here. The steps of Method A are described in detail in this paper and contrasted briefly with the corresponding steps of Method B.
The five steps are sequential with each step adding a new capability to the analysis:
Step 2 allows a visual picture of the path lines,
Step 3 allows for single-phase displacement calculations, and
Step 4 forms the basis fot multiphase displacement calculations done in Step 5.
Stream tubes were first introduced by Higgins and Leighton1 in 1962 for waterflooding using Method A of Table 1. They divided a five-spot along streamlines and mapped a Buckley-Leverett function2 onto the streamlines treating them as I-D conduits. LeBlanc and Caudle3 alternatively used Method B of Table I preferring to calculate pressure over stream function. Martin and Wegner4 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"5 permeability fields. One of these6 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 I, the initial calculation of pressure or stream function is independent of the post processing which occurs in Parts 2 through 5. Additionally, a finite-difference calculation of stream function can easily handle singularities near wells and fractures as will be shown in this paper.
Recently there has been renewed interest in time-of-flight Methods7,8 which are classified here as Method B. These have revived primarily because they provide an inexpensive solution for a true 3D problem.
