"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 (including numerical models).
The purpose of this work is to provide a direct solution mechanism for the case of time-dependent real gas flow which uses an approach that combines the so-called average pressure approximation (a convolution for the right-hand-side nonlinearity) and the Laplace transformation. For reference, 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 a direct solution scheme 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.
Mireles and Blasingame [Mireles and Blasingame (2003)] developed 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. More importantly, they proposed a new approach that uses pseudopressure to linearize the spatial portion of the diffusivity equation (i.e., the left-handside (LHS)) as was done traditionally, but for the right-handside (RHS), (i.e., the "time" portion) Mireles and Blasingame used a special convolution formulation to account for the pressure-dependent, non-linear term. Consequently, the Mireles and Blasingame semi-analytical solution eliminates the use of pseudotime for this case.
Although being rigorous, the Mireles and Blasingame solution relies on evaluation of the non-linear term based on the average reservoir pressure predicted from material balance. They did not assess the nature and applicability of the average pressure approximation (APA), but exhaustively validated the APA approach using numerical simulation for the case of a constant rate inner boundary condition. The effort of Mireles and Blasingame should be considered to be an empirical demonstration of validity of the APA for all (realistic) values of pressure.
Again, the need for such a (direct) solution arises in the analysis of both gas well test data and gas well production data — where both analyses have traditionally used approximate methods such as the pressure or pressure-squared methods, [Rawlins and Schellhardt (1935), Aronofsky and Jenkins (1954)] or rigorous, but tedious pseudovariables [Al-Hussainy et al (1966) and Agarwal (1979)].