Analytical Solution of Stresses Around a Depleting Reservoir Above a Hard Rock Layer with Application to Seismic Hazard Analysis

Abstract

Abstract Surface displacements resulting from reservoir compaction owing to hydrocarbon production are often computed using the nucleus-of-strain concept proposed by Geertsma. This approach enables fast computations for arbitrarilyshaped reservoirs. It was later extended by Van Opstal to calculate surface displacements for compacting reservoirs in the presence of a rigid basement. Later authors further extended Van Opstal’s method to compute displacements, strains and stresses in the 3D subsurface (not just at surface), but those methods were still semi-numerical which significantly hampered computation speed for realistic reservoirs. Here we propose a ‘truly analytical’ solution of displacements, strains and stresses for a compacting reservoir over a rigid basement in the 3D subsurface. Moreover, we propose a method to enable reliable computation of these quantities close to reservoir centres. The approach is applied to Seismic Hazard Analysis (SHA) in depleting (cooling) reservoirs for geometries where no analytical solutions for stress changes at the fault are available. Our results show excellent agreement with earlier publications for compacting reservoirs over a rigid basement. Moreover, our method of applying the nucleus-of-strain concept in combination with subdividing the depleting (cooling) reservoir into thin sub-reservoirs yields stress change profiles along intersecting faults that compare well with exact analytical solutions (which are available for 2D reservoirs). Subsequently, our method was applied to more complicated (non-2D) reservoir shapes, in particular, a tilted fault with nonzero offset intersecting a cylindricallyshaped depleted (cooled) area at an arbitrary place. This is important for the SHA of geothermal doublets, where the cooled zones around the injectors generally have cylindrical shapes. Our results show that for many different geometries, the computed 2D profile of the Shear Capacity Utilisation (SCU) at the fault surface can be reasonably well approximated by a simple 1D SCU along the fault dip based on an analytical model. Furthermore, we show that the presence of a rigid basement has a limited impact on depletion-(cooling-)induced SCU for intermediately dipping faults. However, for faults with low or high dip the presence of a rigid basement does have an impact on SCU.