This paper presents a systematic method for the direct estima-tion of gas-in-place and reserves using only gas production data. The fundamental concept used in this work is "quadra-tic" rate-cumulative relation, which is given as:
Where we define the decline constant, Di, for this case as:
The basis of our method is the development and application of several rate-time/cumulative production-time plotting func-tions which are derived using Eq. 1 (see Appendix A). Eq. 1 is itself derived from a rigorous coupling of the gas material balance equation for a volumetric dry gas reservoir and a modified version of the stabilized gas flow equation for a gas well producing at a constant bottomhole pressure. The proposed procedure is designed to simultaneously match the independent plotting functions (developed from Eq. 1), where we note that each function exhibits unique characteristics (e.g., a straight-line trend) which can be used as an extrapolation mechanism for estimating gas reserves.
In this work we utilize a "spreadsheet" approach in which the data and plotting functions are linked by "global" model parameters, thus ensuring consistent evaluations. We verify our methodology using a simulated case and we then illustrate the application of this method using a field case. Although theoretical considerations limit the rigorous use of the pro-posed methodology to reservoir pressures less than 6,000 psia, we have successfully applied of our method in practice for reservoir pressures as high as 10,000 psia.
The classic works of Arps and Fetkovich2 illustrate the analysis of well performance data using empirically-derived exponential, harmonic, and hyperbolic functions. Although Arps' work is completely empirical (see derivation in Appen-dix B), this work does provide us with a family of rate-time and cumulative production-time relations that are valid (at least in a practical sense) for a variety of producing conditions.
In fact, the Arps' equations continue to enjoy widespread use in the upstream petroleum industry, particularly for production predictions and for estimating reserves from production decline behavior. However, the motivation for this work (i.e., our new reserves estimation methods for gas wells) is our observation that the Arps' relations can yield inconsistent results (unreliable matches, poor extrapolations, etc.). We do not conclude the hyperbolic relations have no utility, but rather, we find that the "quadratic cumulative" relations derived in this work tend to be more consistent, and provide better results for a wide variety of gas reservoir conditions.