The Effect of Vertical Fractures on Transient Pressure Behavior of Wells
Authors: Scott J.O. 
Published: December 2007

Abstract
Transient pressure behavior of wells intersected by a single vertical fracture has been examined by means of a heat flow analogy. Results are correlated in terms of dimensionless pressure change and dimensionless transient time. The resulting curves are found to lie between the dimensionless curves for a cylindrical well and a line source when effective well radius is expressed as one-fourth of the total fracture length. Results are also shown for various degrees of afterflow.
Use of the curves provides a means of determining reservoir permeability, wellbore radius, and thereby fracture length, even when the final slope of the transient curve is not obtained. Superposition of curves may be used to confirm existence of vertical fractures in the reservoir adjacent to the well or to indicate a damaged zone on the fracture face.

Introduction
Hydraulic fracturing has become a standard well completion practice for the purpose of increasing flow. Results are normally gauged in terms of some sort of "stabilized" flow rate established before and after the treatment. This procedure may be misleading because pressures in or around the well do not become "stabilized" during the test period.
Transient pressure tests on fractured wells can overcome the objections to stabilized flow tests by indicating when a stable pressure gradient is reached. In addition, the transient pressure tests can be used to determine formation flow capacity and effective wellbore radius. The effective wellbore radius may be used as a measure of the success of the fracturing operation.
The slope and position of the transient pressure-log time curve may be used to determine the flow capacity and wellbore size after sufficient lapse of time so that the pressure gradient in the vicinity of the fractures becomes constant. However, in tight reservoirs or in short tests this condition may not be reached. Presence of the fractures in tight reservoirs may cause a constantly increasing slope to the transient pressure curve. If this slope is picked too soon, formation flow capacity will appear high and the fracture size will appear too small.
This paper is intended to clarify the early transient behavior of vertically fractured wells. It shows how formation flow capacity and wellbore size can be determined even though the maximum slope of the transient pressure curve is not obtained during the test.

BASIC THEORY
Development of the techniques of transient pressure analysis for wells has been summarized by Perrine. He reviews the various definitions of wellbore effects such as skin effect, productivity ratio, damage factor and condition ratio.In the present paper, use of these terms is avoided by defining an effective wellbore size. For the fractured well this effective wellbore size is larger than the actual wellbore. Assumptions made are constant flow rate at the well, constant fluid compressibility, constant fluid mobility and uniform porosity and formation thickness. From the basic differential equations of flow, two different solutions may be obtained. The first, the line source solution, is commonly used in transient pressure analysis. It expresses the change in pressure at any radius due to withdrawal or injection of fluid at the line source from an infinite reservoir at uniform initial pressure. This equation, in oil field units, is
(1)
or, after a long time:
(2)
The second solution frequently used in transient pressure analysis is based on a finite reservoir. In this case, flow between the reservoir boundary and the well has occurred long enough for a pressure gradient constant in time to exist. Shut-in of the well results in pressure change in the well which, after a sufficiently long time, exhibits a straight-line behavior when plotted against the log of time. For a constant pressure at a circular, concentric drainage boundary with radius, the difference in well pressure and drainage pressure in the straight-line portion is approximately:
(3)
Expressing the initial pressure difference at steady flow as
(4)
the resulting change in well pressure by combining Eqs. 3 and 4 is
(5)
Comparison of Eq. 5 with Eq. 2 shows that they are identical if in either case represents the change in well pressure following the interruption of the stabilized reservoir pressure condition.
JPT
P. 1365^