Radially Symmetric Thermo-Poroelasticity Problems for a Solid Cylinder

Many standard mathematical techniques for solving partial differential equations in applied mathematics can be used to explain at least the simplest experimental results pertaining to geomechanics and to test the accuracy of the computational techniques used in poroelasticity. Thus, it is often possible to solve poroelasticity problems for geometries more complex than a one-dimensional column, examples of which can include a sphere, a cylinder or an ellipsoid. A limitation of the one-dimensional formulation also stems from the fact that manifestations of effects that are particular to poroelasticity cannot be observed unless a two- or three-dimensional formulation is used. Such an example is the Mandel–Cryer effect of amplification of the pore fluid pressure in a radially loaded sphere or a cylinder (Cryer, 1963; Mason et al., 1991; Detournay and Cheng, 1993), or in a poroelastic parallelepiped compressed between two rigid plates (Mandel, 1950; Abousleiman et al., 1996). In this chapter, we examine the thermo-hydro-mechanical problem related to a poroelastic cylinder on the lateral surface of which a non-zero temperature change is prescribed while the fluid pressure and the radial stress are kept at zero. The end faces of the cylinder remain insulated for axial fluid flow and heat transfer. Thus, the fluid flow and heat transfer in such a cylinder occur only in the radial direction. The effect of the mechanical loading of this cylinder can also be considered – it can include, for example, applied axial strain/stress and radial stress applied to the lateral surface. Taking advantage of the linearity of the problem, for the sake of brevity, it is sufficient at first to prescribe zero values for the mechanical loading; for example, zero radial stress on the lateral surface and zero axial strain. It will be shown how the solution of the given thermo-hydro-mechanical problem, with specified non-zero temperature change on the boundary, can be reduced to a problem of prescribed uniform temperature change that does not vary over time. In turn, the last problem can be reduced to a hydro-mechanical problem of the applied radial stress and applied axial strain (or applied axial stress) if certain replacements in coefficients of the solution are performed.