The Application of the Lognormal Cumulative Distribution Function (LCDF) for Time-Rate Interpretation and Forecasting of Wells in Uncon-Ventional Reservoirs

The primary objective of this work is to present a new DCA model adapted from the classic Lognormal Cumulative Distribution Function (LCDF) (NIST [2012], Wikipedia [2023]), where the base result is given by:

(Equation)

The proposed LCDF-DCA model was proposed by the first author of this work by subtracting the LCDF relation from unity (1), and then multiplying by a maximum scale value (i.e., the qi,LCDF parameter), this yields:

(Equation)

Or in its more compact form: [using erfc(x) instead of erf(x)]

(Equation)

Where:

q_i,LCDF = Initial rate (Vol/D)

erf = Error function (Abramowitz and Stegun [1972]) (dimensionless)

erfc = Complimentary error function (Abramowitz and Stegun [1972]) (dimensionless)

μ_LCDF = Model parameter (μ = median LCDF parameter relation) (ln[t])

σ_LCDF = Model parameter (σ = standard deviation for LCDF parameter relation) (dimensionless)

The simplicity (and robustness) of the q_LCDF(t) model suggests that this could be a very effective DCA model, which could serve as a compliment or a supplement to the existing family of "distribution function" DCA models (i.e., the stretched exponential/power-law exponential model, the Logistical Growth Model, and the Weibull model).

As a methodology, the "qDbQ" plot is used to compare and contrast behavior between various model and data functions. The qDbQ plot is a log-log plot with rate [q(t)] and cumulative [Q(t)] plotted on the left-hand scale and the decline parameter [D(t)] and the decline exponent [b(t)] are plotted on the right-hand axis. The model rate function [q(t)] is the basis function, and all of the other functions are computed from q(t) — the D(t) and b(t) functions are computed using analytical relations derived from the q_LCDF(t) model (and validated using high-precision numerical calculations). Unfortunately, the form of the q_LCDF(t) model cannot be integrated analytically, and we are left only with numerical integration — but again, this is performed with high-precision methods (as were the derivatives required for the numerical validations of the D(t) and b(t) functions). In short, we are highly confident of our ability to compute the q_LCDF(t) model and its auxiliary model functions (i.e., the Q(t), D(t), and b(t) functions). In addition to the qDbQ functions, we also use the q’, q_avg, and q_avg′ functions on the qDbQ plots for the LCDF-only matches.

In terms of applications, of testing/applying the new q_LCDF(t) model, we provide 13 example cases — 1 tight gas case (East Texas), 1 Wolfcamp (TX) oil well case, 1 Appalachian gas well case, 3 Eagle Ford (TX) oil well cases, 5 cases from the SPE Data Repository (these are anonymous data donations for shale gas and shale oil wells), and 2 historical test cases from the petroleum literature. For all cases, the Lognormal Cumulative Distribution Function (LCDF), the Modified-Hyperbolic (MH), and the Power-Law Exponential (PLE) DCA models were successfully applied using the qDbQ (log-log) plot approach. The LCDF model generally lies "between" the MH and PLE models in terms of its comparative performance in a visual sense — however; for several cases the LCDF model yielded superior matches for the D(t) and b(t) parameter function trends [as well as for q(t) and Q(t)], indicating that the form of the LCDF relation may be advantageous in general applications for forecasting and estimation of EUR for shale wells.