Transformational decomposition (TD) method for compressible fluid flow simulations
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A new method, the Transformational Decomposition (TD) method, is developed for the solution of the Partial Differential Equations (PDE's) of single-phase, compressible liquid flow through porous media. The major advantage of the TD method is that it eliminates the need for time discretization, and significantly reduces space discretization, yielding a solution semi-analytical in time and analytical in space. There are two stages in the TD method: a Decomposition stage and a Reconstitution stage. In the Decomposition stage the original PDE is decomposed by using a Laplace transform for time, and successive levels of finite integral transforms for space. Each level of finite integral transform eliminates one active dimension, until a small set of algebraic equations remain. The original PDE is thus decomposed into much simpler algebraic equations, for which solutions are obtained in the transformed space. In the Reconstitution stage, solutions in space and time are obtained by applying successive levels of inverse transforms. In contrast to traditional numerical techniques, the TD method requires no discretization of time and only a very coarse space discretization for stability and accuracy. The TD method is tested against results from one- and two-dimensional test cases obtained from a standard Finite Difference (FD) simulator, as well as from analytical models. The TD method may significantly reduce the computer memory requirements because discretization in time is not needed, and a very coarse grid-corresponding to inhomogeneous regions - suffices for the space discretization. Execution times may be substantially reduced because smaller matrices are inverted in the TD method, and solutions are obtained at the desired points in space and time only, while in standard numerical methods solutions are necessary at all of the points of the discretized time and space domains.