Productivity of Vertically Fractured Wells Prior to Stabilized Flow
Authors: Morse Richard A. , Von Gonten W. Douglas 
Published: December 2007

Using numerical simulation, a systematic study has been made of the productivity of vertically fractured wells from the start of production productivity of vertically fractured wells from the start of production until the attainment of stabilized flow. Productivity data are presented for wide ranges of fracture lengths, reservoir conditions, and fluid properties. Results show that for low-permeability reservoirs, well properties. Results show that for low-permeability reservoirs, well productivity varies greatly with time. productivity varies greatly with time.

Introduction
Several authors have presented papers on the effects of hydraulic fracturing on well productivity. Most of this work has been concerned with the effects of fracturing on "stabilized" or "semisteady state" productivity. Prats et al. investigated analytically productivity. Prats et al. investigated analytically the effects of vertical fractures of infinite flow capacity on well productivity as a function of time. Their results showed a significant variation in productivity over the range of dimensionless times productivity over the range of dimensionless times (tD) from 0.01 to 0.5. Productivity for one set of reservoir conditions showed a variation of about eightfold over this range of dimensionless time. Morse and Holditch indicated fractured well productivities varying by factors in excess of 10 productivities varying by factors in excess of 10 over a real-time range of a few weeks or months under some reservoir conditions.
The objective of this work was to make a systematic study of both constant rate and constant pressure well productivities for a range of reservoir pressure well productivities for a range of reservoir rock and fluid properties and for vertical fractures of various lengths.

Constant Rate Results
We noted that the work of Russell and Truitt was presented in sufficient detail to allow calculation presented in sufficient detail to allow calculation of productivity indexdefined here as qo/(p - pw) as a function of dimensionless time. Details of this calculation are shown in the Appendix. Fig. 1 shows the results of such calculations from the data of Russell and Truitt. This figure shows the calculated well productivity index ratio Jf/Js vs tD for various ratios of fracture radius to drainage radius.
To show the relationship between relative productivity (Jf/Js), real time, and rock permeability, productivity (Jf/Js), real time, and rock permeability, Fig. 2 was calculated from Fig. 1 for a given set of reservoir rock and fluid properties using the dimensionless time equation
A fracture of 330 ft (Xf/Xe = 0.5) and a 40-acre square drainage area were assumed. The constant value ofc mu used to construct Fig. 2 could represent a rock porosity of 0.20 and either of the following:
(1) gas with viscosity of 0.01 cp andcompressibility of 10(-3).
(2) oil with viscosity of 1.0 cp andcompressibility of 10(-5).
In other words, Fig. 2 represents the relative-productivity vs real-time relationship for a typical gas or undersaturated oil reservoir.
These results show that for these broadly typical conditions the time necessary for stabilized flow conditions may vary from 7 days to 7,000 days for reservoir permeabilities from 10 md to 0.01 md. Also, it is readily permeabilities from 10 md to 0.01 md. Also, it is readily apparent that in this permeability range a short-time well productivity measured in the first 10 days of production productivity measured in the first 10 days of production can differ from the stabilized productivity by a factor of 10 or more.
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