Summary The nucleus-of-strain method to calculate displacements resulting from a temperature or pressure change at one point (“nucleus-of-strain”) within an elastic solid was originally proposed by Goodier. In later years, this method gained wide popularity in applications of computing surface subsidence owing to depleted reservoirs. The method includes a correction term to account for the presence of a rigid base below the depleted reservoir. Its advantage is that it allows fast computation of subsidence for arbitrarily shaped reservoirs by simply adding the contributions of all different depleted nuclei of strain. In more recent years, the nucleus-of-strain method was extended such that displacements, strains, and stresses everywhere in the 3D subsurface could also be computed. However, in this case, the correction term for the rigid base below the reservoir had to be evaluated numerically, thereby significantly slowing down the computations. In addition, singularities around the reservoir centers led to inaccuracies for points in and close to the reservoir. The latter drawback was recently overcome by 3D analytical integration around those singularities for parallelepiped (rectangular cuboid) reservoirs. In this paper, we present a fully analytical formulation for the rigid-base correction in the calculation of displacements, strains, and stresses everywhere in the 3D subsurface. This analytical formulation results in an acceleration by almost two orders of magnitude with respect to the previous numerical formulation. This offers clear advantages for cases where the depleted or cooled zone has an arbitrary shape and must be represented by many different nuclei of strain. Furthermore, we present an alternative 3D analytical integration of singularities around reservoir centers for radial reservoirs. This can offer an advantage for radially depleted or cooled reservoir areas, such as in geothermal applications where cooled zones often have a radial shape. Our model has been applied to seismic hazard analysis (SHA) around a tilted fault intersecting a depleted (cooled) reservoir with radial shape and with a rigid base below it. This application has resulted in a number of new fundamental insights into the stability of faults intersecting depleted (cooled) reservoirs: (1) Stress concentrations along the fault plane owing to reservoir depletion (cooling) are not only present at the vertical boundaries of the areas where the reservoir parts on either side of the fault overlap, but also at the boundaries of the horizontal reservoir extent; (2) the presence of a rigid base below the reservoir has a limited impact on fault stability for intermediately dipping faults, but a potentially non-negligible impact on fault stability for low or high dip; and (3) the stability of faults intersecting depleted (cooled) reservoirs with radial shape can be adequately described by the stability of faults intersecting 2D depleted (cooled) reservoirs of the same size. The latter insight implies that previously published simple 2D fault stability methods can be reliably used for routine SHA analysis of radially shaped depleted (cooled) reservoirs.
Paul J. van den Hoek (Thu,) studied this question.