A numerical study on explicit vs implicit time integration of the Vermeer-Neher constitutive model

In the oil and gas industry, subsidence modelling aims to predict the deformation of the ground surface induced by hydrocarbon withdrawal from underground reservoir rocks. In reason of environmental and operational implications associated to subsidence, energy companies make a firm commitment to providing accurate estimates and reliable forecasts. This goal can be achieved through numerical simulations with advanced constitutive models accounting for soil/rock plasticity and creep, such as the model proposed by Vermeer and Neher in 1999. Several different implementations of this model have been proposed in the literature, including explicit and semi-implicit time integrations, but a clean and consistent fully implicit formulation is still missing. In this work, in the line of the classical approach for the modified Cam-Clay model, a fully implicit backward-difference integration is proposed and validated. The derivation of the consistent stiffness matrix is also described in detail, together with its validation strategy. The model has also been implemented in a commercial code for finite element analysis through a user-defined material subroutine. We show the expected advantages of the implicit formulation in terms of stability with respect to the explicit formulation. The examples include studies at material point level and at field scale for a case study of subsidence above a synthetic gas reservoir.