Application of Pseudotime to Buildup Test Analysis of Low-Permeability Gas Wells With Long-Duration Wellbore Storage Distortion
Authors: Lee W. John , Holditch Stephen A. 
Published: December 2007

Summary
This paper presents an improved method of analyzing pressure transient tests in low-permeability gas wells using the recently proposed pseudo time concept. The method is particularly helpful in providing less ambiguous type-curve analysis in gas wells with large pressure drawdowns.

Introduction
The purpose of this paper is to present an improved method of analyzing pressure transient tests in gas wells. A fundamental problem in analyzing these tests has been that analysis techniques have been based on methods derived from solutions to the diffusivity equation describing flow of slightly compressible liquids. The liquid equation is linear, but the corresponding equation describing gas flow is not linear. This fundamental difference has continued to cause problems and misinterpretation.
One important step in adapting solutions of the liquid-flow problem to gas flow was introduction of the pseudo pressure concept.1 Replacement of pressure by this variable almost (but not completely) linearizes the gas flow equation. At least two problems remain:In tests with wellbore storage distortion of early test data, no published type curves2,3 provide much assistance in test analysis because all these curves are based on solutions to the liquid-flow problem. More specifically, the liquid-flow solution characterizes wellbore storage distortion with a wellbore constant, C, which is constant throughout the test. In a gas well, the wellbore storage constant C, is the product of wellbore volume, Vwb, and compressibility of gas in the wellbore, cwb.Equation 1When there has been a drawdown from, say, 5,000 to 50 psia in a flow period, the wellbore storage constant during a subsequent buildup test will change by two orders of magnitude during the test. (Large drawdowns of this magnitude are not unusual in closed-chamber drill stem tests and in other tests in tight gas reservoirs.)Large drawdowns also cause inaccuracies remaining in the partially linearized flow equation to grow, even in situations in which wellbore storage distortion ceases at early times. These inaccuracies can cause errors of more than 100% in type-curve-derived fracture length estimates, for example.
In a recent paper 4 Agarwal proposed an intuitive pseudo time, ta, defined asEquation 2
Agarwal found empirically that buildup tests in tight gas reservoirs yielded much more accurate formation and fracture properties when ta replaced t as a plotting function in type-curve analysis. No theoretical basis for this conclusion was provided, and the problem of changing wellbore storage constant was not presented.
This paper presents a theoretical analysis showing that, under certain conditions, pseudo time effectively linearizes the flow equation for gas. It also shows that a plot of pseudo pressure changes vs. elapsed pseudo time should fit a type curve developed for a slightly compressible liquid, with unchanging wellbore storage constant in some cases. These theoretical conclusions are supported by example tests generated on a computer reservoir simulator and by field examples. The paper also presents an example calculation showing how the proposed techniques are applied.