Analytical Solutions For Transient Thermoelastic Stress Fields Around A Borehole During Fluid Injection Into Permeable Media
Authors: Hojka K. , Dusseault M.B. , Bogobowicz A.D. 
Published: March 2010

Abstract
An approximate analytical transient time solution has been developed for temperature and stress fields around a circular borehole during constant rate fluid injection into a permeable porous medium.
The solution implies that significant reduction or increase of tangential (σθ) during cold or hot fluid injection may occur, alld this has applications in borehole stability problems or induced fracturing. This solution may also be used in thermal enhanced oil recovery schemes where hot fluids are injected, in cases of cold fluid injection for waste fluid disposal, or in cases of geothermal reservoir circulation for heat extraction.
The transient temperature field is solved separately for the fluid and the rock. This solution is achieved from energy conservation by coupling convective heat flow carried by an injected fluid from a borehole, and local heat conduction between a fluid boundary layer and a solid grain. First, temperature is solved in radial coordinates by taking into account radial fluid flow divergence; then, the transient time pressure distribution is calculated from the Theis solution.
Once the temperature and fluid pressure distributions have been determined, the stresses around a borehole are calculated taking into account both poroelastic and thermoelastic rock response. It is shown that during fluid injection, σθ on the borehole wall may change, and shear failure or hydraulic fracture may be triggered. Also, stresses are changed farther from the borehole because of fluid and heat propagation into the porous medium.

Introduction
As conventional reservoirs are depleted, new oil recovery methods are deployed in oil production management. Thermal methods are among the more popular enhanced oil recovery methods in viscous oils. For instance, in steam-flooding, hot fluid injection is used, and in fire-flooding, combustion heat is employed to lower heavy oil viscosity and increase its mobility. On the other hand, cold fluid injection may occur during waste water disposal, or may be deliberately used to enhance rock fracturing, thus increasing its permeability. Other cases where borehole behaviour may be affected by thermoelastic stresses include:In drilling, the base of the hole may be cooled as much as 10 °C to 20 °C by the mud;As the mud warms going up the annulus, it may heat upper formations by as much as 10 °C to 20 °C;In water floods, waste process water injection, or geothermal projects, major reductions in stress develop after long-term cool water injection, opening joints and facilitating fracture; and,In gas production, the cooling effect because of gas expansion near the borehole may cause significant thermoelastic stress reductions.
Temperature changes in the rock induce thermoelastic stresses because of thermal strains, and these stresses may lead to rupture or shear. However, if elastoplastic processes and nonlinear deformation moduli effects are assumed negligible, stresses may be calculated from equations for a fluid-saturated, poroelastic solid by incorporating thermoelastic rock volume changes. The equations for poroelastic theory were developed by Biot (1941, 1955, 1956). Schiffman (1971) extended Biot's theory to include thermal effects.