This paper provides the theoretical and practical basis for application of multi-segment Arps production decline models, particularly for multi-fractured horizontal wells used to develop ultra-low permeability resources. Two- and three-segment Arps models have been used in the industry for production forecasting, but the application is usually based on empirical observations and intuitive assumptions. This paper provides the basis for dividing production history of these wells into at least four segments.
We first examined rigorous solutions to governing flow equations for idealized conditions and found the basis to expect (1) early transient linear flow with b ≡ 2; (2) a transition flow regime between transient and boundary-dominated flow (BDF) regimes with continuously changing b about one log-cycle in duration; and (3) BDF with b ≡ 0 for an incompressible fluid. We examined the basis for the Fetkovich type curve, which include BDF stems for compressible fluids with b values ranging from 0 to 1, and found evidence that, under most circumstances, b values should range from 0.4 to 0.5 for gas wells and 0.3 to 0.4 for depletion-drive oil wells. Durations of these flow regimes can be identified using log-log plots of pressure-normalized rate vs. time (preferable) or simply rate vs. time when pressure data are not available. Analysis of the Arps hyperbolic decline model indicates that straight lines with slopes = 1/b are expected during times with unchanging b.
We found that, for multi-fractured horizontal wells, at least four distinct flow regimes should ultimately appear in practice and are related to depth-of-investigation considerations: (1) an early ramp-up in production; (2) a transient flow regime which can last for years with essentially constant b, often near 2; (3) a transition flow regime, lasting for over a log cycle in time and with continuously changing b; and (4) BDF, with essentially constant b, and 0.3 < b < 0.5 in most cases. Additional segments may arise because of changed operational conditions or flow into the stimulated reservoir volume (SRV) from the unstimulated matrix. There is no reason to expect a terminal b of zero except for the rare case of an incompressible fluid.
For forecasting production from wells in resource plays, we should include the minimum of four flow regimes, even when only transient flow has been observed in history, and we should forecast appropriate durations of flow regimes and times at which they expected to begin and end based on considerations discussed in the paper. These considerations are fundamentally important for forecasting of individual wells and for construction of typical well production profiles (TWPs, akatype wells or type curves).