Response Of A Closed Well-Reservoir System To Stress Induced By Earth Tides
Authors: Arditty Patricia C. , Ramey Henry J. , Nur Amos M. 
Published: December 2007

Abstract
The gravitational attraction between the sun, moon, and the earth gives birth to oceanic tides which are visible and sometimes large, and to earth tides whose amplitudes are small due to the low compressibility of the earth compared to that of water. Tidal effects are observed quite often in open pits, and also in closed well reservoir systems. The purpose of this study was:to develop a theory which describes the pressure variations in closed well-reservoir systems pressure variations in closed well-reservoir systems caused by the earth tides,to study the important parameters which determine the amplitude of the parameters which determine the amplitude of the response, andto inspect different sets of field data for gas and oil reservoir systems for which accurate pressure information is available. pressure information is available.
A new expression for the pressure induced at the bore hole by a periodic tidal stress was derived. The important parameters for the amplitude response of a closed well-reservoir system are the permeability, porosity, and the rock and fluid compressibilities. porosity, and the rock and fluid compressibilities

Introduction
The gravitational attraction between the sun, moon, and the earth gives birth to oceanic tides which are visible and sometimes large, and to earth tides whose amplitudes are small due to the low compressibility of the earth compared to that of water. The lunar-solar attraction of the earth generates a state of stress on the earth's surface which induces aradial deformation of the earth. The maximum value of this strain or deformation is around 36 cm (14 in).Tidal effects are observed quite often in open pits. Bredehoeft et al., Bodvarsson, and Robinson and Bellhave shown the existence of a relation between rock characteristics and the amplitude of the response of an open well aquifer system. Johnson described the behavior of such a system in detail. It was observed that tidal phenomena also exist in closed well reservoir systems.
Sinusoidal water level variations (thus formation pressures) observed in open wells are directly related pressures) observed in open wells are directly related earth tide phenomena. Furthermore, it is believed that the amplitude of earth tide pressure changes is related to the characteristics of the formation and to the fluid contained in the formation. Many technical papers describe this kind of dependence for open papers describe this kind of dependence for open well-aquifer systems for shallow wells, while little has been said about deep closed well-reservoir systems.
The solution for a closed well-reservoir system may be prepared from a modification of Bodvarsson'stheory, which explains the mechanism for the open well reservoir case. We will present a solution for the closed well-reservoir problem, and establish the validity of the theory by analyzing field data from different rock and fluid formation types. It will be helpful to review pertinent stress-strain theory and earth tide theory to introduce the new closed well solution.

STRESS-STRAIN THEORY AND THE EARTH TIDE PHENOMENA
The gravitational attraction between the earth and the moon generates a state of stress on the earth'ssurface which induces a radial deformation of the earth. To understand the mechanism of the earth'sdeformation, let us review some of the basic principles of stress-strain theory. principles of stress-strain theory.

Stress-Strain Theory
A stress,, is a force applied to a surface area. It is defined as:
(1)
This stress causes a deformation of the solid, or a strain, epsilon. In the following, we will consider the surface to be the earth surface. The earth is assumed to be a porous, elastic and isotropic medium, which reacts linearly to a stress. This stress is considered positive when compressive, and negative when tensile. The displacement and strain will be considered positive when responding to a positive stress, and negative when responding to a negative stress.