Application of convolution and average pressure approximation for solving non-linear flow problems. constant pressure inner boundary condition for gas flow
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The accurate description of fluid flow through porous media allows an engineer to properly analyze past behavior and predict future reservoir performance. In particular, appropriate mathematical models which describe fluid flow through porous media can be applied to well test and production data analysis. Such applications result in estimating important reservoir properties such as formation permeability, skin-factor, reservoir size, etc. "Real gas" flow problems (i.e., problems where the gas properties are specifically taken as implicit functions of pressure, temperature, and composition) are particularly challenging because the diffusivity equation for the "real gas" flow case is strongly non-linear. Whereas different methods exist which allow us to approximate the solution of the real gas diffusivity equation, all of these approximate methods have limitations. Whether in terms of limited applicability (say a specific pressure range), or due to the relative complexity (e.g., iterative character of the solution), each of the existing approximate solutions does have disadvantages. The purpose of this work is to provide a solution mechanism for the case of timedependent real gas flow which contains as few "limitations" as possible. In this work, we provide an approach which combines the so-called average pressure approximation, a convolution for the right-hand-side non-linearity, and the Laplace transformation (original concept was put forth by Mireles and Blasingame). Mireles and Blasingame used a similar scheme to solve the real gas flow problem conditioned by the constant rate inner boundary condition. In this work we provide solution schemes to solve the constant pressure inner boundary condition problem. Our new semi-analytical solution was developed and implemented in the form of a direct (non-iterative) numerical procedure and successfully verified against numerical simulation. Our work shows that while the validity of this approach does have its own assumptions (in particular, referencing the right-hand-side non-linearity to average reservoir pressure (similar to Mireles and Blasingame)), these assumptions are proved to be much less restrictive than those required by existing methods of solution for this problem. We believe that the accuracy of the proposed solution makes ituniversally applicable for gas reservoir engineering. This suggestion is based on the fact that no pseudotime formulation is used. We note that there are pseudotime implementations for this problem, but we also note that pseudotime requires a priori knowledge of the pressure distribution in the reservoir or iteration on gas-in-place. Our new approach has no such restrictions. In order to determine limits of validity of the proposed approach (i.e., the limitations imposed by the underlining assumptions), we discuss the nature of the average pressure approximation (which is the basis for this work). And, in order to prove the universal applicability of this approach, we have also applied this methodology to resolve the time-dependent inner boundary condition for real gas flow in reservoirs.
average pressure approximation
Zhakupov, Mansur (2003). Application of convolution and average pressure approximation for solving non-linear flow problems. constant pressure inner boundary condition for gas flow. Master's thesis, Texas A&M University. Texas A&M University. Available electronically from