Numerical modeling of volcanic ground deformation

The aim of this work is to study natural processes which take place within the Earth crust and have connections with volcanic activity. When focusing on geological risk associated to volcanoes, it is known that the ground deformation and gravity changes are two indicators of volcanic activity, furthermore precursors of eruptions (see [3] and [5]-[9]). The modeling of the interaction between displacements and gravity change due to internal sources has been developed by several authors. This model is a coupled system of linear parabolic (for the deformations) and elliptic PD equations (for potential gravity change) (see for instance [3] and [5]-[9]). We present the numerical approximation of such a coupled model by means of the finite element method. Sección en el CEDYA 2011: AN 1 Elastic-gravitational model In early eighties Rundle developed the techniques needed for calculation of displacements and gravity changes due to internal sources of strain in layered linear elastic and elastic-gravitational media (see [8] and [9]). The elasticity problem studies relation between body forces and stress. Equilibrium equations represent balance forces under the medium. It describes how displacements change inside of it. Displacement vector, ui, is defined by ui = xi − xi, where x is the initial state and x′ deformation state. We define Cauchy stress tensor as follow: eij = 1 2 (ui,j + uj,i). (1) Comma means partial differential: ui,j = ∂ui ∂xj . Stress tensor is symmetrical. Stress is related to deformations by Hooke’s law. This relation can be expressed by: σij = λδijekk + 2μeij . (2) being δ Kronecker delta and λ and μ Lamé parameters. We consider fi as body force per mass unit and ρfi the force per volume unit. Therefore, relation between deformations and stress tensor is Hooke’s law: