Application of Singular Value Decomposition to Determine Streamline Distribution for Unit Mobility Ratio Displacement
Authors: Marathe R.V. , Shenoy U.V. , Khilar K.C. 
Published: March 2010

Abstract
A new method is proposed to obtain streamline distribution in a horizontalplane of a homogeneous reservoir. The method is a generalization of theclassical method of images used to obtain the potential and velocitydistribution in a symmetric pattern flood. However, it is extremely difficultto use the classical method of images, as shapes of reservoirs and locations ofwells are generally arbitrary in nature. Further, methods based on, finitedifference techniques are difficult to implement, because they requiresubstantial post-processing for front tracking and cannot handle singularitiesnear the well.
In the proposed method, a large number of image wells (N) are arbitrarilylocated outside the reservoir boundary, which can be either sealing? type oruniform pressure (flow potential type). A large number of points M, (M&gt;N)are selected along the reservoir boundary and the appropriate boundaryconditions are written (no flow or isopotential). This leads to M linear, equations In N unknowns, which can be conveniently solved, using the Singular Value Decomposition (SVD) technique for the unknown rates-of the image wells. The iso-potentials, streamlines and single-fluid fronts can then be obtainedusing, the principle of superposition.

Introduction
Knowledge of fluid front locations at various times plays a key role indesigning a suitable development plan for better exploitation of oilreservoirs. Depending on the nature of the drive mechanism, the problem may beone of suggesting a suitable pattern of wells for pressure maintenance in adepletion-drive pool or of suggesting locations of production wells in awater-drive reservoir to achieve a uniform movement of water-oil contact tomaximize areal sweep.
The streamtube model was first introduced by Higgins and Leighton(1). They described a computational technique to predict twophase flow through streamtubes generated from a potentiometric model for afive-spot pattern. The streamtubes (which lie between single-fluid streamlines)are divided into cells of equal volume, and average penneabilities andgeometrical shape factors determined for each cell are used to compute a Buckley-Leverett displacement. The overall performance is obtained by summingup the production from all streamtubes. Morel Seytoux2 demonstratedthe use of conformal mapping techniques to obtain potential distribution insymmetry elements of staggered line drive and direct line drive patternsassuming unit-mobility ratio displacement. Abbaszadeh-Dehghani and Brigham(3) presented use of conformal mapping technique to obtainpotential and velocity distributions for symmetric pattern floods assuming unitmobility ratio displacements. In addition, they performed tracer dispersioncalculations in individual streamtubes as the injected tracer slug propagatesin the reservoir. Le Blanc and Caudle(4) suggested use of a methodof images technique to handle arbitrary shaped reservoirs, However. theirmethod involves a trial and error approach.
Martin and Wegner(5) utilized a finite difference model instreamtube-isobar coordinate system with streamtube locations beingperiodically updated. Their paper presented justification to the assumptionmade by Higgins and Leighton, and Le Blanc and Caudle that waterfloodperformance can be calculated by holding streamlines constant as the floodfront progresses.