This work presents the development of two new rate-time relations which are based on self-growth limiting time-cumulative relations. This self-limiting feature provides an inherent upper limit on ultimate reserves and eliminates the need for a terminal decline component as is required in other time-rate relations. Another inherent advantage of this approach is that these new models introduce EUR as a regression parameter instead of using the "intercept rate" as the general regression parameter (as is the case in the Arps time-rate relations and most subsequent models).
As validation of these two new relations we employ synthetic solutions (i.e., reservoir simulation) as well as field performance data taken from a well-documented tight gas case and from two North American shale gas cases. As a summary statement, the new relations tend to be more "conservative" estimators (like the power-law exponential and other statistical relations (e.g., the Logistic Growth Model)) and less like the Arps' hyperbolic family of relations. In general, the new models match all of the cases reasonably well, but (as noted), the forecasted production and estimated ultimate recovery (EUR) extrapolations tend to be conservative. Unfortunately, the new models do not provide any direct diagnostic characteristics where the parameters in these relations could be estimated directly (e.g., as in the case of using the slope and/or intercept of a straight-line trend).
Lastly, in this work we do provide a series of "time-cumulative" plotting functions in an attempt to provide data diagnostics which are less affected by data noise inherent in production data. These relations appear to be potentially useful — however; a concern remains regarding the introduction of new data diagnostic functions as the "Arps'" functions (D(t) and b(t)) are the standard variables used in practice and it is unlikely that industry practice will embrace new functions which do not provide significant advantages over the Arps relations.