An Improved Pseudo-Time for Gas Reservoirs With Significant Transient Flow
Authors: Anderson D.M. , Mattar L. 
Published: March 2010

Abstract
The use of semi-analytic methods for correcting flow equations to accommodate changing gas properties with pressure has become increasingly common. It is a mainstay of modern production decline analysis, as well as gas deliverability forecasting. The use of pseudo-time is one method which enables a time-based correction of gas properties, honouring the gas material balance within the time-based flow equation. By using pseudo-time, the analytical well/reservoir models, derived for the liquid case (slightly compressible fluid), can be modified for gas by re-evaluating the gas properties as the reservoir pressure depletes. These gas correction procedures are well documented in the literature. Also well documented is the iterative nature of the gas properties correction methods, as original gas-in-place is a required input into the equations.
The pseudo-time correction is based on the average reservoir pressure and works very well for boundary dominated flow. However, when transient flow prevails, the pseudo-time concept is not valid and its use can create anomalous responses. This will occur in low permeability systems or in reservoirs with irregular shapes, especially where some of the boundaries are very distant from the well.
The semi-analytic gas correction has a ‘representative pressure’ at its root, which, in the existing models, is always the average reservoir pressure. We propose a straightforward modification to the determination of this pressure as follows. The representative pressure ought to be based on a ‘radius of investigation’ or ‘region of influence’ (in the case of non-radial systems), rather than the average reservoir pressure. In the case of a depleting system, the representative pressure would be the same as the average reservoir pressure. The following paper outlines the proposed procedure and illustrates its advantages over the existing method by using synthetic and field data examples.

Introduction

Background
Literature on the derivation and usage of pseudo-time is prevalent(21,32). The definition that will be used in this paper is shown below:
Equation (1) (Available In Full Paper)
The above is used in the pseudo-steady-state equation for gas, which is at the core of most modern production decline analysis methods. It is also used in analytical well/reservoir models, whose conventional formulations are only valid for slightly compressible fluids with constant properties over a given pressure range. These models enjoy widespread usage for both history matching and forecasting, and their inclusion of pseudo-time for gas reservoirs is vital.
To illustrate the value of pseudo-time, let us take the simple case of a vertical well in the centre of a circular gas reservoir. We will assume constant rate production and pseudo-steady-state conditions. Thus, the model that describes the pressure response at the well can be reduced to the pseudo-steady-state equation for gas(5 3).
Equation (2) (Available In Full Paper)
The ƒ(t) in Equation (2) is the chosen time function. Figure 1 shows the pressure response plotted against time for two cases: ƒ(t) = time (t) and ƒ(t) = pseudo-time (ta). Also compared is the numerical solution using the same input parameters. Upon comparison of the solutions, it is clear that pseudo-time has a significant impact on the flow equation for gas.