Applications of Rate Normalization for Gas Performance Analysis (Part VI)
“All models are wrong, but some are useful” (George Box, 1976)
💠Introduction
The preceding articles in this series explored the concept of normalization and its application in analyzing a variable-rate/variable-pressure history. It was shown that normalization is equivalent to rigorous superposition. By applying this simple yet powerful concept, a complex production history can be analyzed without compromising rigor. However, the linearized form of the diffusivity equation obtained for liquids is not directly applicable to compressible fluids —gases— due to the absence of a material balance correction that accounts for changes of the μct product as depletion takes place. This article covers the following aspects that are central to gas flow analysis,
✅Pseudo Time Transformation
✅Equivalent Liquid Behavior Formulation
💠Pseudo Time Transformation
Unlike liquids, the viscosity and compressibility of gases can change as pressure decreases during depletion. This effect introduces non-linearity into the governing flow equation. Therefore, to linearize the diffusivity equation and apply liquid-based solutions for the Cartesian analysis of boundary-dominated flow in a gas well, an integral transform is required. Fraim & Wattenbarger (1987) proposed such a transform called normalized time,
The normalized time, , is evaluated at the average reservoir pressure as opposed to the definitions given by Agarwal (1979) and Lee & Holditch (1982), which use the wellbore pressure, and are intended for the analysis of transient data
💠Equivalent Liquid Formulation
⏩Gas Flow Equation - Constant Back Pressure
By combining the material balance form for gas, including compressibility effects (Ramagost & Farshad, 1981),
With the pseudo-pressure transform given by (Meunier, Kabir, & Wittmann, 1987),
And the productivity index definition, from the PSS formulation,
It can be shown that the following expression can be obtained
or, equivalently,
Therefore, under boundary-dominated conditions with constant back-pressure, the flow rate exhibits an exponential decline when assessed using normalized time, consistent with the results established by liquid-phase analyses. The rate-normalized time data, when matched using the Arps-Fetkovich Type curve, should fit the stem b = 0, corresponding to an exponential decline. This key finding demonstrates that, with an appropriate transformation, gaseous systems can be analyzed, similar to liquid systems. It is important to note that calculating normalized time, tn, is an iterative process; the value of G, which is initially unknown, must be determined to compute reservoir pressure, which is subsequently required to evaluate the μc product.
⏩Gas Flow Equation - Variable Back Pressure
While the exponential solution applies to the constant back-pressure scenario, the more general case of a variable pressure/variable rate can be addressed using a similar treatment. For a smooth varying rate schedule, a liquid-equivalent form for post-transient gas flow is given by,
With
The pseudo material balance time for gas, ta,mb, is (Blasingame & Lee, 1988; Palacio & Blasingame, 1993),
The adjusted cumulative production, Gpa is given by (Blasingame and Lee 1988),
Henceforth, during boundary-dominated conditions, a Cartesian plot of the Δp(pwf)/q against ta,mb yields a straight line with intercept bpss,g , the inverse of the productivity index, and slope m_Δp/q, just like the liquid case (See Part I)
The adjusted material balance time is computed through numerical integration with the following approximation,
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The in-place, G, is calculated from the slope of the Cartesian plot,
The kh product can be determined from
Note that this computation requires a knowledge of the shape of the drainage area, as well as the skin factor estimation
✋The full derivation of these equations is available upon request
💠Computational Procedure
To compute ta,mb, the PVT properties must be estimated at the prevailing reservoir pressure. The latter is determined from the material balance equation, which in turn requires the value of G. However, G is the unknown value being solved for. This creates a circular problem that must be tackled through an iterative process. The iterative scheme can be set up as an optimization problem defined by
A numerical root-finding algorithm, like the Newton-Raphson method, is used to solve for G. The process is illustrated in the flowchart below
💠Application
Fetkovich et. al. (1987) presented the case of a stimulated low-permeability gas well, which has been maintained at a constant pressure for the majority of its operational life.
The log-log diagnostic plot shows the distinctive features of post-transient conditions:
The figure below illustrates that using the material balance time effectively linearizes the data on the Cartesian plot. Except for the early time period (t < 25000 hrs.), the plot using the conventional time is mostly non-linear, rendering the liquid solution invalid.
The in-place is determined from the Cartesian analysis using the iterative procedure described in the previous section. After convergence is achieved, the slope of the Cartesian plot is used to estimate the in-place,
A skin factor of -5.5 from a transient test is available; hence, permeability is obtained as follows,
The observed production data exhibit a strong correlation with the Arps-Fetkovich b = 0 stem. This behavior is consistent with theoretical expectations for a well producing under constant back-pressure when normalized time is used as the time function. This further confirms the onset of exponential decline within the reservoir system
💠Takeaways
✅ Linearization is Mandatory: Standard Cartesian analysis fails in gas systems because μc variation can trigger non-linear artifacts during depletion.
✅ Consistency with Liquid Solutions: When correctly normalized, gas flow data aligns with liquid analytical solutions, enabling the use of established Caretesian, type-curve, and decline-curve techniques.
✅ Pseudo-Time vs. Material Balance Time: While transient analysis uses wellbore-pressure-based transforms, BDF analysis requires transforms based on average reservoir pressure.
✅ The Iterative Trap: Calculating OGIP from gas flow data is inherently circular; robust analysis requires a numerical root-finding algorithm like Newton-Raphson.
💠References
A very useful linked article, especially for REs dealing with gas wells as it provides refresher to what to look out for when moving from oil to gas well/reservoir. BTW, not sure what George Box really meant in his quote; All models are wrong, but some are useful - despite being wrong? i believe some are useful despite being wrong, though not useable without appropriate corrections. For instance, here, linear oil models are not useable in gas but are useful as they give an idea how to modify them to make them useable in gas too. I'm amazed by past researchers (and present ones too!) who utilized powerful diagnostics feature of pressure derivative plots from oil to gas by using pseudo pressure and pseudo time. Indeed, hats off to all such researchers mentioned in this article.
Strong technical insight and very well explained. Great work Sir!
Excellent work Adolfo. Thank you for sharing your knowledge.
ADOLFO D'WINDT Certainly "𝘼𝙡𝙡 𝙢𝙤𝙙𝙚𝙡𝙨 𝙖𝙧𝙚 𝙬𝙧𝙤𝙣𝙜, 𝙗𝙪𝙩 𝙨𝙤𝙢𝙚 𝙖𝙧𝙚 𝙪𝙨𝙚𝙛𝙪𝙡" (𝙂𝙚𝙤𝙧𝙜𝙚 𝘽𝙤𝙭, 1976). Thanks for sharing your expert knowledge
This is useful, thanks for sharing Adolfo