Further Analysis Of Production-Performance Graphs
Authors: Purvis R.A. 
Published: March 2010

Abstract
Techniques for interpreting rate-ratio-cumulative graphs for harmonic decline are described. These graphical techniques are a continuation of those for rate-ratio-time graphs for exponential decline described by the author in a recent JCPT paper. As before, the slope of various graphs is interdependent.
Some specialty graphs are presented where the slope of the rate or ratio graphs is the analytical derivative of a cumulative production variable. Because of the inherent smoothing associated with cumulative data it is often advantageous to use cumulative graphs to define, or at least confirm, the slope of interdependent graphs.
The techniques are applicable to both single-well and aggregated data for a lease, project, or pool. The examples emphasize the identification of the factors contributing to the observed production decline.

Introduction
Routine decline-curve analyses use time and cumulative production as the independent variables. Time is the most convenient independent variable because extrapolations of rate-time graphs can be directly used for production forecasts and lease evaluations. Although time is the most convenient independent variable, rate-cumulative graphs have the advantage that an extrapolation to the economic limit provides a direct estimate of the ultimate recovery. The rate-cumulative graph also shows the sensitivity of recovery to the economic limit. Moreover, shut-in times for workovers, pressure surveys, market conditions, etc. do not detract from graphical continuity. For harmonic decline, it is well known that log (rate) vs cumulative production exhibits a linear relationship. In addition to rate and ratio graphs, several other dependent variables have been proposed in the literature to facilitate interpretation of cumulative production graphs. Arps(1) demonstrated the utility of a few variables for pools subject to different drive mechanisms. For water-drive pools, Arps suggested a graph of log (fo) vs Qo. The ultimate recovery is obtained by extrapolating the log (fo) graph to a limiting oil cut. For solution-gas-drive pools, he reasoned that while only a fraction of the oil is recovered, most of the solution gas is produced. Therefore, the solution gas-in-place provides a logical end point for a graph of log (Qg) vs Qo. For pools with advancing oil-water contacts, Arps proposed extrapolating a graph of contact elevation vs Qo. The ultimate recovery is then estimated from the elevation of the top of the producing zone in the well or lease.
Ershaghi and Omoregie(2) proposed a method for extrapolation of water flood performance. The method assumes that recovery is controlled by a fractional flow curve based on a linear log (kro/kru) vs Sw relationship. A graph of log [(l/fo) – l/fo] vs Qo is extrapolated to a limiting oil cut. The method was tested(3) by applying it to some field data and to the results from a reservoir simulator. Startzman and Wu(4) compared the method to the conventional log (WOR) vs Qo graph and concluded that both methods appeared equally valid for high water-oil ratios (WOR).
Pool geometry is also an important consideration when forecasting future performance. For pools subject to vertical drives, Purvis(5) presented some schematic rate-cumulative graphs to illustrate the influence of a few simple geometries on oil rate.