A New Method for Computing Pseudo-Time for Real Gas Flow Using the Material Balance Equation
Authors: Rahman N.M.A. , Mattar L. , Zaoral K. 
Published: March 2010

Abstract
Analytical solutions in well-test analysis are oriented towards fluids with a constant viscosity and compressibility in a porous medium with a constant porosity. To use these solutions in gas-flow situations, one needs to apply the pseudo-pressure and pseudo-time transformations. Thus, the non-linear diffusivity equation for gas flow is transformed into a linear one, allowing one to use the solutions for fluids with constant compressibility, viscosity, and porosity. Although the computation of pseudopressure is reasonably accurate, the conventional computation of pseudo-time by direct integration of the compressibility-viscosity function over time can result in significant errors when modelling gas reservoirs with residual fluid saturation, rock compressibility, and a large degree of depletion. Errors in calculating the pseudo-time result in substantial errors in the material balance, which has an adverse impact on reservoir modelling and production forecasting.
In this study, a new method for computing pseudo-time is presented. This method is based on the material balance equation that considers the rock and fluid compressibility. This formulation honours the material balance equation in all situations. Examples are presented to show that the problems with computing pseudo-time, using the traditional definition, can be resolved when the new method is used. Accurate computations of pseudo-time allow one to use the solutions for fluids with constant compressibility and viscosity for modelling and forecasting gas production.

Introduction
Analytical solutions are generally used to analyze and model well test and production data. However, these solutions have been developed for fluids with a constant viscosity and compressibility and for formations with a constant porosity. The governing diffusivity equation and its boundary conditions are linear when expressed in terms of pressure, space, and time variables. The analytical solutions to these equations are reasonably accurate for the liquid-flow situations. In contrast, the diffusivity equation for gas flow and its boundary conditions are non-linear when expressed in terms of pressure, space, and time variables. As no analytical solutions to these non-linear equations are available, one needs to apply pseudo-pressure and pseudo-time transformations in order to use the analytical solutions for liquid flow in gas-flow situations. This approach introduces two variables in the diffusivity equation for gas flow?pseudo-pressure (Ψ) as the dependent variable, and pseudo-time (ta) as an independent variable. As a result, the diffusivity equation for gas flow is transformed into a linear one, allowing one to use the slightly compressible fluid (liquid) solutions. In this study, we are considering a single-phase gas flow situation in the presence of residual fluid saturation and a compressible formation. Here gas is the only mobile phase, while oil and water phases are immobile, if there are any. The pseudo-variables (pseudo pressure and pseudo-time) can be defined as:
Equation (Available In Full Paper)
The rationale for defining the above pseudo-variables is demonstrated in Appendix A. Martin(3) made the first systematic attempt to define the total system compressibility in multi-phase conditions, neglecting the rock compressibility. Later, Ramey(4) followed Martin's lead and included the formation compressibility in defining the total system compressibility, ct, as: