| Title | Quaternions to Model Non-Isothermal Poroelastic Phenomena |
|---|---|
| Authors | Mario-César Suárez Arriaga and Fernando Samaniego Verduzco |
| Year | 2011 |
| Conference | Stanford Geothermal Workshop |
| Keywords | quaternions, thermo-poroelasticity, EGS |
| Abstract | Quaternions are hypercomplex quantities in four dimensions (q0, q1, q2, q3); they are a generalization of the classical two-dimensional complex numbers (x, y) = x + i y, where i 2 = - 1. Quaternions satisfy all the algebraic properties of the traditional complex numbers except that their multiplication law is not commutative. They 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. In this paper we present for the first time, an original application of quaternions to model non-isothermal poroelasticity. It is well known that the presence of a moving fluid in elastic porous rocks modifies their mechanical response. Poroelasticity explains how the fluid inside the pores bears a portion of the total load supported by the rock. The other part of the load is supported by the skeleton. The total deformation of geothermal rocks originates from lithostatic compression, pore pressure and thermal stresses. The introduction of the volumetric Gibbs free enthalpy as a fundamental thermodynamic potential, allows to include the thermal stresses directly. We introduce the thermoporoelastic theory by defining poroelastic coefficients coupling four quaternions: two for the bulk rock, one for the fluid and one for the total thermal expansion/contraction. The need of the fourth dimension appears naturally and permits to extend the theory of linear elasticity to geothermal rocks, taking into account the effect of both the fluid pressure and the temperature changes. Isothermal poroelasticity contains 16 different coefficients, but only four of them are independent. Introducing four volumetric thermal dilation coefficients, one for the fluid and three for the skeleton, a complete set of parameters for non-isothermal rocks are obtained. We show how the set of quaternions we are introducing is equivalent to a 4D-tensorial poroelastic model. In other words, the thermoporoelastic functions mimic the algebraic structure of a non- commutative field. To illustrate the practical use of this formulation some applications are outlined: full deduction of the classical Biot |