Unsteady-State Pressure Distributions Created by a Well With a Single Horizontal Fracture, Partial Penetration, or Restricted Entry
Authors: Gringarten Alain C. , Ramey Henry J. 
Published: December 2007

Abstract
Although there have been many studies on unsteady behavior of wells with vertical fractures, and although there was at one time a controversy concerning the occurrence of horizontal or vertical fractures as a result of hydraulic fracturing, to date there bas been published no study of the unsteady behavior of a well containing a horizontal fracture. This is particularly surprising because such a study might have indicated significant differences between the performance of wells with horizontal fractures and those with vertical fractures. The purpose of this study was to fill that existing gap in knowledge of fractured-well behavior.An analytical solution was developed by means of the concept of instantaneous sources and Green's functions. The analytical solution modeled the behavior of constant-rate production from a well containing a single, horizontal fracture of finite thickness at any position within a producing interval in an infinitely large reservoir with impermeable upper and lower boundaries. This general solution also contained solutions for the cases ofa single, plane (zero thickness) horizontal fracture,partial penetration of the producing formation, andlimited flow entry throughout a producing interval.
Although those are interesting solutions, the main purpose of this study was to investigate the horizontal fracture case. The analytical solution for this case was evaluated by computer to produce tables of dimensionless pressures vs dimensionless times sufficient for well-test analysis purposes. A careful analysis of the general solution for a horizontal fracture indicated the existence of four different flow periods. It appears that during the first period all production originates within the fracture, causing a typical storage-controlled period. This period is followed by a period of vertical, linear flow. There Then follows a transitional period, after which flow appears essentially radial. During the last period, the pressure is The same as that created by a line-source well with a skin effect. The skin effect is independent of time, but does depend upon the position of the pressure point It was found that there is a radius of influence beyond which flow is essentially radial for all times. Approximating solutions and appropriate time limits for approximate solutions were derived.

Introduction
Hydraulic fracturing has been used for improving well productivity for the last 20 years and is generally recognized as a major development in well-completion technology. There was considerable discussion in the early 1950's about the orientation and the number of fractures created by this type of well stimulation. It is now generally agreed that a vertical fracture will result if the least principal stress in the formation is horizontal, whereas a horizontal fracture will be created if the least principal stress is vertical. Further, data collected and reported by Zemanek et al. shows that hydraulic fracturing usually results in one vertical fracture, the plane of which includes the axis of the wellbore. This conclusion appears widely held today. Thus, most studies of the flow behavior for fractured wells consider vertical fractures only.However, the existence of horizontal fractures has been paved in some cases, and various authors have considered them. The steady-state behavior of horizontally fractured wells has been studied numerically by Hartsock and Warren. Their model assumed the reservoir to be homogeneous, of constant thickness, of anisotropic permeabilities, and completely penetrated by a well of small radius. A single, horizontal, symmetrical fracture of negligible thickness and finite conductivity was located at the center of the formation. Radial flow was assumed beyond a critical radius four times as large as be fracture radius, and there was no flow across the drainage radius. The only flow into the well itself was through the fracture.
SPEJ
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