A Fast Method for Computing Load Induced Stress in the Earth

Abstract

Summary Deformation of the Earth by a large load is usually modelled by the loading of a semi-infinite elastic half space. Solution of this Boussinesq problem can be achieved by numerical integration of the equations for deformation due to a point load, or by integral transform methods. The solution for the deflections is found by Fourier transforms using the fast Fourier transform algorithm for determining the reservoir topography spectrum. After using a well-known solution due to Love on the average load, the stresses and displacements can be calculated at the surface. The method is illustrated by applying it to the Kariba Reservoir, and it is found the method is up to twenty times more efficient than numerical integration. Hydrostatic loading of the Earth is often, but not always associated with an increase in regional seismic activity, (Gough & Gough 1970a, b; Lane 1971; Ellis er al. 1974; Howells 1972; Rotlie 1970). This paper is concerned with a method of calculating the stress in the Earth due to such loading. This loading is only one of several causes of induced seismic activity but other causes will not be examined here. Gough (1969) and Gough & Gough (1970a) presented two papers in which they examined the stress pattern below lakes. They represented the load due to the bathymetry of Lake Kariba as a system of point forces and computed the stress and displacement within the lithosphere by an application of the solution to Boussinesq’s problem. (See for instance Landau & Lifshitz 1957.) They acknowledge that their method is very expensive in computer time, and we present here a more efficient method for this computation. Our numerical results reproduce theirs but we can extend our computations to zero depth whereas Gough & Gough were limited by their method of representing the load. Our theoretical formulation also shows promise of generalization to layered, porous, water filled media. Landau & Lifshitz (1957, pp. 26-29), as well as numerous other texts, show that for a semi-infinite elastic half-space of Poisson’s ratio 0, and Young’s modulus E, the equilibrium equation for small deformation is