A Simple Method for Estimating Well Productivity

Abstract

Abstract We present a simple method for estimating the well productivity of a natural completion. The method gives quick, accurate estimates of productivity that agree well with established finite-element calculations. We show that three nondimensional groups - determined by the major governing parameters: penetration length, shot density, perforation tunnel diameter, permeability anisotropy, wellbore damage length and perforating damage - control most of the functional dependence of well productivity. One practical application of the method is in selecting the best perforating system from a suite of alternatives. The analysis is also useful in determining economical trade-offs between marginal increases in productivity and contemplated improvements in any of the perforating parameters (perforation length, shot density and tunnel diameter). Introduction Researchers have been devising various methods for estimating well productivity from a perforated formation for many years. Most methods assume that the flow in the permeable formation can be modeled by potential theory, that is, as a solution to Laplace's equation, subject to particular boundary conditions. An oft-quoted study is that of Tariq, who used a finite-element method to solve numerically the well flow over a wide range of perforating and formation parameters. These results were later reduced to a series of analytical approximations by Karakas and Tariq that now serve as the basis for two commercially available computer codes that are in common use today. Although useful in estimating well productivity and in assessing trade-offs between different gun systems, the computer analyses sometimes tend to obscure insight to the relative importance of the various competing parameters. Nondimensional analysis can often be of great use in these situations by grouping together the most important parameters to reveal the underlining functional dependencies. And that is the purpose of this study. Basis of the Method The method assumes that the dominant variables determining productivity are: perforation length (P), shot density (N), diameter of the perforation tunnel (d), anisotropy or ratio of horizontal permeability to vertical permeability of the formation (a), diameter of the wellbore (D), length of local wellbore damage (L), and the damage caused by the perforating jet (bc). The method assumes that there is no appreciable difference in productivity caused by the phasing of the gun, as long as the perforations are distributed along a spiral pattern. Both the wellbore damage and the perforating damage (the local impairment to permeability caused by the perforating jet) are initially assumed to be zero. Later on, we show that these effects can also be included as part of the analysis. No Damage. Before considering the general case of wellbore and perforating damage, consider the simpler situation of no wellbore damage (L = 0) and no perforating damage (bc = 1). We define the productivity ratio (PR) in the conventional way as the ratio of the actual flow through the perforations, Q, to that of the flow, QD, into an open hole having a diameter equal to the wellbore, D, as (1) We also define a maximum productivity ratio, PR, based on the flow, Q, into the wellbore of an idealized gun having penetration, P, and an infinite shot density (i.e., N). Such a gun - if there were such a thing - would enlarge the wellbore to a diameter (D + 2P). P. 57^