Modifications to Production Increase Calculations for a Hydraulically Fractured Well
Authors: Soliman M.Y. 
Published: December 2007

Soliman, M.Y., SPE, Halliburton Services

Discussion
Production increase of a well as a result of a vertical Production increase of a well as a result of a vertical fracture is affected by propped length, propped height. and width and permeability of the fracture. If a steady-state condition of production is assumed, the production increase may he defined as the ratio of production rate after fracturing to the rate before fracturing at the same pressure differential. The assumed steady-state condition pressure differential. The assumed steady-state condition requires constant pressure at the outer and inner boundaries. These boundary conditions are employed in models used by Tinsley et al. Prats, and Mao. Since the model presented by Tinsley et al. considers the effect of the fracture height, it is more general than the other two models. Because of this generality the graphs obtained using that model are used widely in petroleum industry. However, the analysis presented in petroleum industry. However, the analysis presented in their paper is not complete. Consequently. the graphs given in that paper are not correct. The purpose of this paper is to present a correct version of these graphs. The new graphs are compared with curves given by Prats and Mao.

Modification to Tinsley et al. Method
To show the nature of the correction to the Tinsley et al. method, the mathematical analysis presented in their paper is summarized briefly. paper is summarized briefly. The "fracture relative capacity" plotted on the x-axis of Tinsley curves is developed by using Eqs. 2 and 5 in Ref. 1:
1/ ...................(1)
....................(2)
In Tinsley's study, the resistance of the fractured system (gamma fs). the radial system (gamma i), and the fracture (gamma f) were measured. Next, gamma i/gamma fs was plotted vs. (gammai/p) Using Eqs. 1 and 2, these values were translated into their physical meaning in petroleum engineering literature. physical meaning in petroleum engineering literature. .........................(3)
...............(4)
Tinsley's curves could he modified by redefining the x and y axes. If gamma i/gamma f is plotted along the x axis (instead of gamma i/p), the x axis will be redefined in the more desired dimensionless form:
=1/2 ...............(5)
This modification will eliminate the need to use a scaling factor in the x axis. It also will shift the curves with respect to the x axis. This shift causes the shape of the curves to coincide with other production increase curves i.e., Prats and Mao. To scale up the model to any general dimension, a scaling factor in the y axis is needed. This scaling factor should account for the change in dimensions of the radial pattern. Eq. 6 describes the scaling factor. pattern. Eq. 6 describes the scaling factor.
= ...................(6)
In this equation, Subscript 1 could he the actual value in the potentiometric model, while Subscript 2 could correspond to any other condition. The model used by Tinsley et al. has an re/rw ratio of 500. Thus, this value is used in the new graphs. A sample of the modified production increase curves is presented in Fig. 1. A complete set of these curves is presented in Fig. 1. A complete set of these curves is available from the author.
JPT
p. 170