Relating In Situ Stresses and Transient Pressure Testing for a Fractured Well
Authors: Ochs Daniel E. , Chen Her-Yuan , Teufel Lawrence W. 
Published: December 2007

Abstract
The purpose of this study is to link the flow-induced rock-stress responses to the transient fluid-pressure for a well with a stationary vertical fracture. The problem is examined within the framework of Biot's theory of poroelasticity. Solutions for fluid-pressure and rock-stress responses are presented for an infinite, isotropic, homogeneous reservoir in a state of plane strain. Results show that the induced changes in vertical stress and horizontal mean stress can be estimated from the fluid pressure change. The induced changes in horizontal normal stresses (perpendicular and parallel to fracture plane) can be represented by two parts:a hydrostatic part (horizontal mean stress) which is proportional to the fluid-pressure change, anda deviatoric part which is equal in magnitude but acts in an opposite manner.
The deviatoric component also represents the difference between the changes in horizontal stresses. This deviatoric component approaches a constant value during the pseudo-radial flow period.

Introduction
A number of problems of current interest to the petroleum industry require understanding of the evolution of stress state in the reservoir, e.g., reservoir compaction/surface subsidence, stress reversal/refracturing, and stress-sensitive permeability. A consistent and coupled consideration of geomechanics and reservoir engineering is essential in these types of problems (see, e.g., Ref. 1). Specifically, a working model consistent in both fluid flow and geomechanical considerations is required to link various fluid-rock information (e.g., flow/storage properties, rock mechanical properties, reservoir fluid-pressure and stress level) measured by different types of techniques (e.g., core, well testing, logging, in situ stress measurement), and to forecast reservoir performance.
In this study, linking pressure transients to the corresponding evolution of in situ stresses is the primary goal. Motivation primarily stems from our interest in extending current well testing technology to the estimation of reservoir mechanical properties and in situ stress level. Other practical issues which are related to and may benefit from this study are: the effect of fracture orientation on well pattern/spacing, altered-stress fracturing and fracture reorientation, and stress-sensitive reservoirs.
The specific problem addressed here is the evolution of fluid pressure and rock stresses induced by production/injection from a well with a vertical fracture. Solutions for fluid pressure surrounding a fractured well are available in the literature. The corresponding stress responses, however, have not been examined in a consistent, compatible, and convenient manner.
The objective of this study, thus, is to present a consistent model which can be used to study the simultaneous evolutions of reservoir fluid-pressure and stresses due to production from a fractured well. This paper will focus more on the theoretical aspects than on the practical applications.

Problem Statements
The problem under consideration is to determine the induced fluid-pressure and rock-stress changes due to production of a slightly compressible fluid from a two-wing, vertical fracture confined within a horizontal permeable layer with impermeable upper and lower boundaries. The reservoir is considered to behave according to the linear poroelastic theory of Biot. Fracture and reservoir properties (permeability, porosity, and mechanical properties) are homogeneous and isotropic.
The sign conventions adopted in this study are that the stress and strain are taken positive in tension whereas fluid pressure is positive in compression.
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