Abstract
We undertake a formal derivation of a linear poro-thermo-elastic system within the framework of quasi-static deformation. This work is based upon the well-known derivation of the quasi-static poroelastic equations (also known as the Biot consolidation model) by homogenization of the fluid-structure interaction at the microscale. We now include energy, which is coupled to the fluid-structure model by using linear thermoelasticity, with the full system transformed to a Lagrangian coordinate system. The resulting upscaled system is similar to the linear poroelastic equations, but with an added conservation of energy equation, fully coupled to the momentum and mass conservation equations. In the end, we obtain a system of equations on the macroscale accounting for the effects of mechanical deformation, heat transfer, and fluid flow within a fully saturated porous material, wherein the coefficients can be explicitly defined in terms of the microstructure of the material. For the heat transfer we consider two different scaling regimes, one where the Péclet number is small, and another where it is unity. We also establish the symmetry and positivity for the homogenized coefficients.
Highlights
The theory of consolidation of soils goes back to the work of Terzaghi (1944) and Biot (1941, 1972, 1977), and since numerous authors have contributed to the field, extending the models to different situations and providing more rigorous results for the equations
Notable contributions are Burridge and Keller (1981), where a formal upscaling leading to the quasistatic Biot-model was undertaken, and the book Sanchez-Palencia (1980) where a rigorous derivation can be found
We omit all superscripts in the variables, subscripts in the differential operators, and introduce a more familiar notation for the coefficients, similar to what is commonly used in the literature on the quasistatic poroelastic equations: α := (|Y f | I − BH ), K := KH, c0 := G H, β := (|Y f | M −UH ), Θ := Θ H as: (Θ H), a0 := M H, A := AH Ξ := Ξ H, b0 := E H, where α is the Biot–Willis constant, c0 is the specific storage coefficient, and A is the effective elastic moduli, containing the elastic coefficients of the porous medium
Summary
The theory of consolidation of soils goes back to the work of Terzaghi (1944) and Biot (1941, 1972, 1977), and since numerous authors have contributed to the field, extending the models to different situations and providing more rigorous results for the equations. We shall focus on a natural system, such as the subsurface, where flow velocity, mechanical strain, and temperature changes are small This allows for linearization of the constitutive laws of thermoelasticity, as well as linearization of the fluid-structure coupling conditions. The homogenization of a similar model problem was undertaken by Lee and Mei (1997), but with a different scaling, and with the fine-scale model defined in terms of Eulerian coordinates This approach leads to relatively strict conditions on the allowable deformations. Our justification for the upscaled model comes from the similarity with the isothermal poroelastic equations, and the analogy to the thermoelasticity equations in mechanics
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
