This work formalizes the Arps' D and b-parameters using a semi-analytical flow relation for a well produced at a constant bottomhole pressure in an HP/HT reservoir. The semi-analytical formulation is given in rate-time and rate-cumulative forms, and the definitions of the D and b-parameters are derived directly from the semi-analytical form.
The rationale for this work is the over-dependence that many analysts place on the Arps' D and b-formulations (i.e., the original hyperbolic rate-time model), tuned to specific (i.e., constant) values of the D and b-parameters. The purpose of this paper is to establish guidelines for reserve estimations and for production extrapolations in general. The relations developed in this work can be applied to a number of different HP/HT tight gas and shale gas field cases — in this work we validate our proposed procedure with a simulated example and we apply the methodology to a fractured gas well in an HP/HT reservoir. A comprehensive workflow is provided, as well as the relevant details/formulations of the analytical flow solution for HP/HT gas cases. We also provide decline curve plots/plotting functions to yield reserve estimates and production extrapolations.
Unconventional gas resources comprise a very important share of the domestic natural gas resource base and offer tremendous potential for future reserve and production growth. Unconventional gas resources — in particular, tight gas sands — are characterized by very-low/low permeability and low porosity reservoir properties. And as the natural gas industry continues to explore for more resources, exploration and development activities extend to much greater depths. These depths exceed 15,000 ft and are approaching to 25,000 ft. The reservoirs in these depths exhibit abnormally high initial pore pressure and temperature gradients resulting in high-pressure/high-temperature (HP/HT) reservoir conditions.
The common practice in the industry in reserves assessment is to use the Arps'  hyperbolic rate decline relations. Similar to conventional gas resources, tight gas sand reserves are frequently (often solely) assessed with Arps' hyperbolic rate decline relations. The Arps rate decline relations are only specifically valid for boundary-dominated flow conditions, and it is often noted that the decline curve parameter, b, should lie between 0 and 1. However, we frequently observe b-parameter values greater than 1 — particularly in tight gas sands at HP/HT reservoir conditions, and it is well established that for cases of b<1, the reservoir is in transient (or transition) flow. It is inappropriate (in a strict reservoir engineering sense) to use the Arps models to estimate reserves for cases where b<1 as this will yield significant errors in reserve estimates — particularly in unconventional resources (e.g., tight gas, shale gas, HP/HT gas reservoirs, etc).
This is not to say that the hyperbolic relation cannot be used as an "interpolation" or "fitting equation" for transient flow data — in fact, this appears to be a common practice in the work of economic reserves practitioners. While empirical fitting of transient flow data using the hyperbolic decline relation is "empirical" at best (i.e., an appropriate reservoir model should be used), our concern lies not in the use of the hyperbolic relation for "fitting" the transient flow data, but rather, in the extrapolation of the transient flow (hyperbolic) trend to estimate reserves. Extrapolation of transient flow data using any empirical model to estimate reserves is an inappropriate practice, and should never be conducted in any capacity other than building possible scenarios — not for estimating reserves!