# Finite volume method for modelling of linear elastic deformation

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## Abstract

The demand for accurate and efficient simulations in order to test the geomechanical effects is a reality for the entire geoscience community. The motivation that arises from that need is the development and the evolution of modelling methods to study these effects. Deep understanding of any problem in fine scale is crucial, especially when it extends to much coarser scales.
In this work the finite volume method (FVM) is used for mechanical modelling of deformation in elastic media. The momentum balance equation is solved as the governing equation for mechanics, assuming linear elasticity for the stress tensor. Here, displacement is mapped onto a vertex-centred grid in three dimensions (3D). A set of eight trilinear basis functions are used to locally interpolate the value of displacement within each grid cube. In the finite volume method, the discretized form of the equations are obtained by integrating the governing equation over control volume surfaces, since in 3D the control volume is a cube. Hence, discretized forms are obtained by considering 24 surfaces, which form between a displacement node and its neighbouring displacement cells. This required extensive derivation. The implementation of the numerical model was carried out by writing a code in MATLAB. Several numerical test cases are presented to demonstrate the capability of this model. In the first place, the consistency of the model is checked through comparison with synthetic analytical solutions, which are compared to the numerical solutions. Furthermore, the simple test case of uniaxial compression, has been carried out with this model, but also compared to the results with a 2D FVM model and a 3D finite element (3D FEM) one. In another test case, ground plain strain subsidence is studied in a real hydrocarbon field with a heterogeneous map for elasticity parameters. It is shown that 3D FVM is in close agreement with 3D FEM in predicting the subsidence due to field depletion. Last but not least, displacement and stresses, in a faulted reservoir in which fluids are injected, are modelled and the results are shown to coincide with a robust analytical solutions for the system. Finally, the aim of this work is to shed some more light on the finite volume method for mechanics and bring it closer to the audience of science.