Application of Convolution Theory for Solving Non-Linear Flow Problems: Gas Flow Systems

In this paper we present the development of a rigorous approach for the solution of non-linear partial differential equations by use of the Laplace transformation — in particular, the convolution theorem for Laplace transforms. While the rigor of this new approach is general, our paper is devoted to the development, verification, and application of this method for the case of real gas flow in porous media. This paper focuses on verification of real gas flow solutions using the results of numerical simulation. The major results of this work are: Development of a general analytical approach in the Laplace domain for solving non-linear partial differential equations. Verification and application of this new approach for the flow of a real gas in porous media. We observe excellent agreement between the numerical and analytical results for the case of a single well produced at a constant rate in a closed reservoir. We conclude that this approach could be adopted as a method of verification for numerical simulation, as well as for analytical modelling of the gas flow case — in particular, for modelling future performance directly and accurately, without numerical simulation or some weak approximation (p or p approximations, etc.). In addition to the specific problem considered in this work, the flow of a real gas in porous media, we believe that it may be possible to extend this work for the development of analytical solutions for multiphase flow. Introduction The primary objective of this work is to develop a closed form Laplace domain solution for the flow of a real gas from a well producing at a constant rate in a bounded circular reservoir. The need for this solution arises in the analysis of gas well test data and gas well production data, where both analyses currently use approximate methods such as the pressure or pressure-squared methods, or rigorous, but tedious pseudovariables (pseudopressure and pseudotime). Our new solution can also be used for predicting reservoir performance as a high-speed simulation device — as opposed to using numerical reservoir simulation solutions. While our new solution is not necessarily "easier" to apply than say the pseudovariables approach, our solution is essentially exact and it can be applied directly in performance prediction, as opposed to implicitly as in the case of the pseudotime approach (in theory, the pressure-dependent properties must be known). Our only "approximation" is the referencing of the time-dependent viscosity-compressibility product to the average reservoir pressure as a function of time, as computed from material balance. This referencing seems not only logical, but also appropriate (particularly when one considers that the pseudotime function also has this type of formulation). We present the solution methodology where we recast the non-linear term of the right-hand-side of the gas diffusivity equation into a unique function of time. We do this without regard as to how to couple the time-pressure relationship, this will come later. We then use the convolution theorem for the Laplace transformation, as well as the definition of the Laplace transformation in order to "transform" our gas diffusivity equation into the Laplace domain. In the Laplace domain, we recognize that the non-linear term is simply a transform function that can be incorporated directly into the "liquid equivalent" form of the Laplace domain solution. The final issue is the resolution of how to sample the non-linear term, and as such, we empirically establish that for the constant rate case, the non-linear term should be sampled at the average reservoir pressure predicted from material balance. This sampling is later shown to yield an essentially exact comparison with the numerical solution and its pressure derivative functions. We also present a comprehensive validation of our convolution theory/Laplace transformation approach by comparing our new solution to the results of a finite-difference reservoir simulation model for a variety of cases of gas reservoir size, initial pressure conditions, and gas properties. This chapter provides conclusive evidence that our new solution yields an essentially exact solution for the bounded circular reservoir case. SPE 84073 Application of Convolution Theory for Solving Non-Linear Flow Problems: Gas Flow Systems T.J. Mireles, BP, and T.A. Blasingame, Texas A&M U. Copyright 2003, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in Denver, Colorado, U.S.A., 5 – 8 October 2003. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s).