Keywords: porous structures, poroelasticity, thermoporoelasticity, finite elements.

Quaternions are hypercomplex quantities in four dimensions that have important applications in aeronautics, space flight dynamics, robotics, control theory, signal processing, and in computer graphics because of their stability and ability to represent three-dimensional rotations [1]. In this paper an original application of quaternions is presented to model non-isothermal poroelastic processes [2]. Poroelasticity explains how the fluid inside the pores bears a portion of the total load supported by rocks and other porous structures. The other part of the load is supported by the skeleton [3]. The Gibbs free enthalpy is a fundamental thermodynamic potential, which allows the inclusion of thermal stresses in non-isothermal porous phenomena [4]. In this context, the need of the fourth dimension appears naturally allowing the theory of linear elasticity to be extended to geothermal rocks, taking into account the effect of both the fluid pressure and the temperature changes. A set of quaternions is introduced, which are equivalent to a four-dimensional poroelastic tensorial model. To illustrate the practical use of this formulation some applications are outlined: full deduction of the classical Biot's theory coupled to thermal stresses, construction of a general, well posed mathematical model to represent the physical behavior of geothermal rocks. Deformations of a non-isothermal reservoir, solved with finite elements, are included. The examples presented suggest the strong influence of temperature changes on the poroelastic strains and on the poroelastic coefficients. The poroelastic deformations are much larger in geothermal reservoirs with higher and variable temperature than in isothermal aquifers. For reservoirs containing cold water, the estimated value of vertical strains is about 1.5x10^-4, while for hot water reservoirs the same strain is about 7.5x10^-4. These results can be applied to geothermal and hydrocarbon reservoirs, to deep aquifers, and to other non-isothermal porous structures.

1

J.B. Kuipers, "Quaternions and Rotation Sequences: a Primer with Applications to Orbits, Aerospace, and Virtual Reality", Princeton University Press, 1999.

2

J. Bundschuh, M.C. Suárez, "Introduction to the Numerical Modelling of Groundwater and Geothermal Systems: Fundamentals of Mass, Energy and Solute Transport in Poroelastic Rocks", CRC Press, Taylor & Francis Group, 2010.

3

M.A. Biot, "General theory of three-dimensional consolidation", Journal of Applied Physics, 12, 20-24, 1941. doi:10.1063/1.1712886

4

O. Coussy, "Mécanique des Milieux Poreux", Editions Technip, Paris, 1991. doi:10.1121/1.402718

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