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The premise of this work is the concept, development, and application of an approximate relation for the material balance of abnormally pressured gas reservoirs. In particular, the proposed approximation is formulated directly from the rigorous material balance for the case of an abnormally pressured gas reservoir. Our "quadratic cumulative production" result is given by: p p p i i G G G G G z p z p of function a as defined is 1 1 where 2(1) The -function is defined by the "cumulative effective compressibility" )] ( [ p e c and represents the influence of abnormal pressure. If we assume =constant (but this does not mean that ) ( p e c is constant), then we obtain the "quadratic" form: G G G G z p z p p p i iand where 2 1  (2) Eq. 2 is the base definition of the "quadratic cumulative production" relation ( is constant; therefore,  and  are constant). We have shown this relation to be an extraordinarily accurate approximation of the rigorous material balance for the case of a dry gas reservoir with abnormal pressure effects. Eq. 2 is suited not only for use as a characteristic model, but also for use as a data analysis mechanism (i.e., in this work we use Eq. 2 to develop a suite of analysis plots, plotting functions, a type curve, etc.). We also address the case of  being a linear function of Gp (as opposed to  being constant) — which results in a formulation where p/z is cubic in terms of Gp. In this work, we provide the following new results:  The "quadratic cumulative production" model for the material balance behavior of abnormally pressured gas reservoir.  A suite of 6 (six) plotting functions based on the p/z-Gp 2 material balance model.  A suite of 4 (four) -Gp performance plots which are used to calibrate the analysis process.  A new type curve pD[=(pi/zi-p/z)/pi/zi] versus GpD[=Gp/G], based on the new p/z-Gp 2 material balance model. Objectives The primary objectives of this work are:  To develop a quadratic formulation of the rigorous material balance for the case of an abnormally pressured gas reservoir in terms of cumulative gas production (derivation is given in Appendix A).  To develop plotting functions for the analysis of reservoir performance behavior based on the quadratic cumulative production formulation of the rigorous material balance for the case of an abnormally pressured gas reservoir. We note that we use 10 (ten) specific data plotting functions as part of this work — others are available, but we favor the 10 functions selected due to consistency and data representation/visualization that these functions provide. These functions are derived in Appendix B.  To develop and validate a dimensionless "type curve" solution based on Eq. 1 and an auxiliary function (i.e., the so-called "pressure integral" of Eq. 1 based on Gp) (Appendix C).  To validate and demonstrate the plotting functions and associated analysis relations using simulated reservoir performance cases, as well as various field performance cases. 2 F.E. Gonzales, D. Ilk, and T.A. Blasingame SPE 114044 Introduction The rigorous material balance for the case of an abnormally pressured gas reservoir was developed by Fetkovich et al [Fetkovich et al (1998)] and is given as:

A Quadratic Cumulative Production Model for the Material Balance of an Abnormally Pressured Gas Reservoir