Finite-Element Analysis of Temperature and Thermal Stresses Induced by Hot Water Injection
Authors: Aktan T. , Ali S.M. Farouq 
Published: December 2007

Abstract
Results of an experimental study of the temperature and pressure dependence of rock elastic moduli were used in a numerical mathematical model that describes the unsteady-state, two-dimensional flow of a single-phase, single-component, slightly compressible hot fluid and that calculates the state of stress in porous media. The numerical formulation was conducted using the finite-element method of triangular elements to discretize the space domain and backward differencing for discretizing the time domain.
Sections of the model dealing with the flow of mass and heat, and stress calculations were tested separately then, the complete model was used to investigate the effect of temperature-dependent Young's moduli and Poisson's ratios on the state of stress and the propagation of thermal and pore stresses with an advancing hot water front in porous media. porous media

Introduction
The finite-element method forms the basis for a versatile analysis procedure applicable to problems in several different fields. Essentially, the finite-element idealization replaces the continuum with a finite number of discrete elements. Geometry of the elements is defined by a set of spatial points (called nodal points) of the system. Heat and fluid-flow problems are analyzed by using interpolation functions for the unknown temperatures and pressures. Thus, the variables within elements are pressures. Thus, the variables within elements are defined completely in terms of the values at nodal points. The isoparametric element concept, where points. The isoparametric element concept, where the geometry and displacements of the element are described in terms of the same parameters and are of the same order, is particularly useful for stress analysis in continuous bodies.
The investigation of thermal stresses may be found helpful when answering questions associated with heated reservoir rocks. Creation of microfactures in formations subjected to elevated temperatures often results in increased permeabilities. A correlation between the thermal stresses induced by these high temperatures and the degree of microfracturing affecting the rock absolute permeability could be useful in understanding fluid flow permeability could be useful in understanding fluid flow in heated reservoirs. It also would be interesting to see if thermal stresses have any effect on the casing failures in steam injection or geothermal wells.
The analysis under consideration involves the determination of pressures, temperatures, and stresses in a porous medium undergoing hot-water injection. The study is complicated further by the transient nature of the variables.

Background
The finite-element method has been used widely for studying stresses in various kinds of structures. The basic concepts of the method, first laid down by Turner et al, essentially are composed of the idealization of continuous bodies by a set of interconnected finite elements with known behavioral characteristics. Recently, this versatile method was applied to diverse kinds of engineering problems, mainly in heat and fluid-flow areas. Desai problems, mainly in heat and fluid-flow areas. Desai and Abel list the fields where the method may be applied.
Visser solved the unsteady-state heat conduction equations, using a variational approach. He solved for temperature distribution, also calculating the associated stresses by determining the displacements occurring in the body due to this temperature field. Later, Zienkiewicz and Parekh used two- and three-dimensional curved isoparametric elements to solve the same equation by using the Galerkin principle.
SPEJ
P. 457