The Transformational Decomposition (TD) Method for Compressible Fluid Flow Simulations

Abstract

A new method, the Transformational Decomposition (lD) method, is developed for the solution of the Partial Differential Equations (PDE's) of single-phase, compressible liquid now 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 lD method: Introduction In transient now through porous media, the Partial Differential Equation (PDE) to be solved is obtained by combining appropriate forms of Darcy's Law and the equation of masS conservation, yielding: a Decomposition stage and a Reconstitution stage. In the Eq. 1 is generally nonlinear, and in all but the siInplest DecompOSition stage the original PDE is decomposed by problems is solved numerically. The basic concept of any using a Laplace transform for time, and successive levels numerical method is the substitution of a set of algebraic of finite integral transforms for space. Each level of finite equations for the original PDE. Instead of solving for the integral transform eliminates one active dimenSion, until a continuous smooth function p(x, y, z, t), the space domain small set of algebraic equations remain. The original PDE (x, y, z) is subdivided in ND subdomains, and the. tinte t is thus oecomposed into much simpler algebraiC equations, is discretized in NT timesteps; NT sets of approximations for which solutions are obtained in the transformed space. p of the solution are obtained at the N D predetermined In the Reconstitution stage, solutions in space and time points in space. A PDE problem with a continuous smooth are obtained by applying succesive levels of inverse transsolution surface is thus reduced to· a set of algebraic forms. In contrast to traditional numerical techniques, the equations, which are easier to solve and provide a solution . lD method requires no discretization of time and only a. arithmetically "close" to the true solution, from which they very coarse space discretization for stability and accuracy. differ by the truncation error e = p _ p. _ The TD method is tested against results from oneand Despite their power and flexibility, numerical solutwo-dimensional test cases obtained from a standard Finite tions have some serious drawbacks. Minimization of the Difference (FD) simulator, as well as from analytical moderror introduced by the numerical approximation of the spaels. The TD method may significantly reduce the computer tial derivatives in the PDE's dictates the discretization of memory requirements because discretization in time is not the space domain into a large number of subdomains at all needed, and a very coarse grid corresponding to inhomoof which solutions must be obtained (whether desired or geneous regions suffices for the space discretization.· Exnot). This increases the execution time requirements and ecution times may be substantially reduced because smaller requires a large amount of computer memory, especially matrices are inverted in the TD method, and solutions are when direct matrix solvers are used. The approximation obtained at the desired points in space and time only, while of the time derivatives in the PDE's is one of the most in standard numerical methods solutions are necessary at important sources of instability and error. Accuracy and all of the points of the discretized time and space domains. stability considerations necessitate a large number of small timesteps between observation times; solutions must be obtained at all these intermediate times, increasing the execu2 THE TRANSFORMATIONAL DECOMPOSITION (TO) METHOD FORCOMPRESSIBLE FLUID FLOW SIMULATIONS SPE 25264 tion times and the roundoff errors. The problem of restriction on the size of 6.t is exacerbated by the nonlinearity of the PDE, which is caused by the pressure dependence of the liquid density and the formation porosity. This necessitates even shorter timesteps, dictates internal iterations within each timestep until a. convergence criterion is met, and adds significantly to the computational load. The Transformational Decomposition (ID) method is a new method which addresses the aforementioned shortcomings of traditional numerical techniques. The TO method was first applied to the solution of the diffusiontype (parabolic) PDE of incompressible flow through porous media 1. It is based on successive integral transforms, and is an extension of the approach used by Goode and Thambynayagam2 in the analysis of pressure response of horizontal wells~ ne major advantage of the 1D method is that it requires no time discretization and a very coarse . space discretization to yield an accurate, stable solution which is semi-analytical in time and analytical in space. In this paper, the 1D method is formulated to address the problem of slightly compressible, single-phase liquid flow through porous media. The mathematical basis of the method is develqped, and its performance is evaluated against analytical solutions and standard FD models. The Transformational Decomposition (TO) Method If gravity is negelected and porosity is considered constant, then Eq. 1 becomes successive levels of integral transforms. The first step in this stage involves the application of the Laplace transform to eliminate the time dependency oCthe original PDE. The resulting equation is then subjected to successive finite integral (for space) transforms. Since virtually all boundaries in petroleum reservoirs are "no-flow" (Le. Dirichlet-type), the finite cosine transform is employed. Each level of finite cosine transform eliminates one active dimension, until .single point equations remain. The original PDE is thus decomposed into much simpler point algebraic equations, for which solutions are obtained in the transformed space. In the Reconstitution stage, solutions in space and time are obtained by applying succesive levels of inverse transforms. The development of the TO method is described in detail in the following sections. Step 1: The Laplace Transform of the PDE_ The Laplace transform of Eq. 3 yields A('I1) = V . [A VW -1] (Cp '11 + 1/8)] = CT [8'11r(O») + ii, (5) where AO is the operator defined in Eq. 5, 8 is the Laplace domain parameter, reO) is the distribution of rat t = 0,