Application of convolution and average pressure approximation for solving non-linear flow problems. constant pressure inner boundary condition for gas flow

摘要

The accurate description of fluid flow through porous media allows an engineer to properly analyze pastbehavior and predict future reservoir performance. In particular, appropriate mathematical models whichdescribe fluid flow through porous media can be applied to well test and production data analysis. Suchapplications 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 implicitfunctions of pressure, temperature, and composition) are particularly challenging because the diffusivityequation for the "real gas" flow case is strongly non-linear. Whereas different methods exist which allowus to approximate the solution of the real gas diffusivity equation, all of these approximate methods havelimitations. Whether in terms of limited applicability (say a specific pressure range), or due to the relativecomplexity (e.g., iterative character of the solution), each of the existing approximate solutions does havedisadvantages. The purpose of this work is to provide a solution mechanism for the case of timedependentreal gas flow which contains as few "limitations" as possible.In this work, we provide an approach which combines the so-called average pressure approximation, aconvolution for the right-hand-side non-linearity, and the Laplace transformation (original concept was putforth by Mireles and Blasingame). Mireles and Blasingame used a similar scheme to solve the real gasflow problem conditioned by the constant rate inner boundary condition. In this work we provide solutionschemes to solve the constant pressure inner boundary condition problem. Our new semi-analyticalsolution was developed and implemented in the form of a direct (non-iterative) numerical procedure andsuccessfully 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 andBlasingame)), these assumptions are proved to be much less restrictive than those required by existingmethods 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 nopseudotime formulation is used. We note that there are pseudotime implementations for this problem, butwe also note that pseudotime requires a priori knowledge of the pressure distribution in the reservoir oriteration 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 theunderlining assumptions), we discuss the nature of the average pressure approximation (which is the basisfor this work). And, in order to prove the universal applicability of this approach, we have also appliedthis methodology to resolve the time-dependent inner boundary condition for real gas flow in reservoirs.